Systems of Linear Equations
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We introduce the central problem of solving a systems of linear equations in several variables, starting with two lines in the plane, and then graduating to three planes in 3 dimensional space. There are three basic possibilities corresponding to the geometric orientations of the lines or planes involved: a unique solution, no solution, or an unbounded (or "infinite") number of solutions.
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hello everyone I'm Norman Wahlburgers
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today we're gonna be looking at systems
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of linear equations which is sort of the
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heart in some sense of linear algebra
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solving linear equations but not just
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one of them several of them and not just
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one variable but several variables okay
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so that's the challenge okay so here is
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the XY plane we have two lines one of
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them is given by the equation X plus 2 y
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equals 2 and the other one is 3x minus y
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equals 3 and what we want to do is we
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want to find this common point of
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intersection or the meet of these two
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lines and that of course is finding a
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point XY that satisfies both the
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equations ok so what we're going to do
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is we're going to sort of eliminate one
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of the variables and find the other
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variable and then we'll substitute that
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back into one of the equations to get
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the first row okay
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so let's call these equation something
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let's call this one number one and this
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one number two let's say and then what
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we're going to do is we're going to take
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some combination of them so that the
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total X's that we get is 0 so we're
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going to eliminate X ok so maybe I'll
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say that once here so we're going to
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eliminate X how do we do that we're
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going to take the combination we're
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going to take the equation two and
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subtract three times the equation of
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wannna
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so we'll have 3x minus x so we're gonna
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get 0 X's when we do that and how many
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Y's are we gonna get so there's - y from
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Equation 2 and then we're gonna subtract
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3 times equation 1 so subtracting 3
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times 2 for a total of minus 7y and on
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the right hand side we have 3 you know
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write that down 3 minus 3 times 2
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ok so altogether that's the equation
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minus 7y equals 3 minus 6 is minus 3 so
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y equals 3
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over seven and then what we do is we can
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substitute that back in maybe go up here
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so therefore X plus 2 times 3 over 7
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equals 2 and that's like x equals 2
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minus 6/7 which is altogether 8/7 okay
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so we get that 8 that X is equal to 8 7
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so this point here we've now found its
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coordinates the x-coordinate is 8 over 7
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and the y-coordinate is 3 over 7 and in
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this case there's a unique solution this
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is the only solution
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so up here this is the case of having a
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unique solution there's only one point
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which lies on both lines all right so
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now let's scroll to a next example here
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we have two lines one of them is given
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by the equation 2x plus 3y equals 6 and
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the other one is given by 4x plus 6y
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equals 16
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now I graph these lines for you and they
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appear to be parallel ok in fact they
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are parallel because the normals we can
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read off from the equations of the lines
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pretty easily so normal to the first
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line is the vector two three and a
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normal to the second line would be the
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vector 4 6 that's how we get the normal
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from the equation we look at the
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coefficients of x and y coefficients of
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x and y and these two vectors 2 3 & 4 6
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are multiples of each other they're
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parallel and so these lines themselves
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are going to be parallel so that tells
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us that we are unlikely to find a joint
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solution to both of these lines we are
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not going to find
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and X Y which lies on both lines but
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let's try anyway so we'll call this one
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one and we'll call this one too and
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we'll go through the same procedure that
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we did before so let's take a Quay ssin
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two and subtract two times equation one
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to eliminate the X's it's our idea okay
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what happens well we get 4x minus 2
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times 2 X 2 that's 0 X how many Y's
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there's 6 y minus 2 times 3y so I guess
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that's 0y and 16 minus 2 times 6 is 4
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now 0x plus 0 y equals fourth that's a
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relatively simple equation because it's
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really the equation 0 equals sort and
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there's no solutions to that doesn't
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matter what you plug in for x and y
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there's no solution so that's the answer
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here there's simply no solution there's
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no common intersection or meet of these
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two lines now third example we have okay
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we have two equations but actually the
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lines geometrically look very similar so
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the equations are 2x minus y equals -2
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and 6x minus 3y equals minus 6 so maybe
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we didn't know we have looked at that we
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don't know that they actually overlap
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because we haven't gone through the
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trouble of drawing a diagram if we just
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work algebraically how can we tell you
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know what's gonna happen so let's do the
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same thing that we did before we'll
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label them say equations 1 in equation 2
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and then we'll try to eliminate let's
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say the X's okay so how we gonna do that
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they're taking the equation 2 and
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subtract 3 times equation 1 so the
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equation 2 minus 3 times equation 1 and
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what does that give us
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well 6x minus 3 times 2x it's gonna be 0
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X and then the 3 minus 3 y minus 3 times
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minus 1 so that's plus 3y so B plus
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zero y equals minus six minus three x
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minus two so it's minus six plus six
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that's equal to zero what we're seeing
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there is that the difference between
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these two equations is really zero
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that's sort of what that's telling us
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these are the same equations so what are
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the solutions to this joint pair of
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equations well it's just all the points
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lying on the line so there's in this
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case an unbounded number of solutions
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you can cook up as many as you want
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number of solutions so any point on
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saint-just equation 2x minus y equals -2
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and how could we actually sort describe
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points like that well one way would be
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to set the state let's set one of the
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variables let's say y equals to a
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parameter let's say lambda okay then the
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equation would become 2x minus lambda
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equals -2 so that X will be lambda minus
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2 all over 2 1/2 so that's 1/2 lambda
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minus 1 so in terms of x and y so we get
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a solution that x and y that vector is
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is 1/2 lambda minus 1 and second
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coordinate lambda this is a
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parameterization of the line on the
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parameterization of all the solutions
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parameterization
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of the solutions so you plug in any
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value of lambda into this expression and
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you're going to get a solution in other
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words a point on the line we do
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recognize here a kind of a parametric
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equation for the line and I remind you
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that we can rewrite this by taking the
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lambdas out so we could write this as
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lambda times one half one plus the
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vector minus one zero and what does that
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tell us that means it's a line through
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the point minus one zero in the
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direction one-half one that makes sense
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there's the point minus one zero and it
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has Direction one half one one half over
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one up so there's a direction vector for
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the line okay so in this case there's an
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unbound number solution so there's three
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possibilities that we found we found an
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unbounded number of solutions that was
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one case over here we found that there
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was no solution in that case and the
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first example there was a unique
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solution so these are the three
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possibilities that basically occur over
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and over again in the subject so as we
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proceed to higher dimensions it's going
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to be the same story in that a system of
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linear equations yes I'll write this
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we'll see that any system of linear
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equations has either any solution or no
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solution or an unbounded or some people
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like to say and
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infinite number of solutions I'll write
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that down reluctantly an infinite number
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of solutions I'm not a big fan of
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infinite sets okay great so one of the
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kind of key drivers of linear algebra is
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to find a technology that will deal with
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the big system of linear equations and
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determine whether we either have a
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unique solution well we have no solution
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or we have an unbounded or infinite
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number of solutions okay
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and of course if there are solutions we
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would like to know what they are so if
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there's a unique solution while we'd
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like to know what it is if there's no
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solution there's nothing to say but if
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there's an unbounded or infinite number
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solutions there's also something to say
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we can attempt to describe describe all
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the solutions in a kind of fashion that
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I was doing up here okay that's a way of
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describing solutions in terms of
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parameters so here we might have some
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parameters to describe the solutions
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okay so now to go further what we want
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to do is we want to go to three
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dimensions okay in 3d we're working in
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let me use x1 x2 x3 little bit more
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sophisticated all right so we have three
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axes x1 x2 x3 and we might imagine that
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we have a number of different equations
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and so I need to remind you a little bit
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about what linear equations represent in
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3-dimensional space so for example
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suppose that we have the equations now
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it's going to be in terms of X 1 X 2 and
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X 3 so X 1 minus X 2 plus X 3 equals say
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1 and maybe 2 X 1 minus X 2 equals 3
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and maybe X 1 minus 2 X 2 plus 2 X 3
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equals -2 so now we have three equations
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in three unknowns and we would like to
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understand what the solutions are so we
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want to describe the points X 1 X 2 X 3
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that satisfy all three of the equations
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now I need to remind you okay recall
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that a single linear equation in your
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equation like say X 1 minus X 2 plus X 3
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equals 1 represents a plane a plane in
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three-dimensional space
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okay that's crucially important and and
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this particular plane so how could we
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sort describe that well one easy way is
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to look at the X 1 X 2 and X 3
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intercepts so when X 2 and X 3 are 0 and
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X 1 has to be 1 so if there's 1 then
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that would be a point on the plane
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if X 1 and X 3 are 0 then X 2 would have
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to be minus 1 so we'd let's put some
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scale here okay so there's minus 1 so
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that would be another point on the plane
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and when X 1 and X 2 are 0 then X 3 is 1
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so that would be another point on the
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plane so altogether the plane ends up
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being the plane that passes through
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these three points okay you might know
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it draw it like that but it actually
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extends this two-dimensional fashion so
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it's a plane in three-dimensional space
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okay now it's a little bit onerous to in
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general to draw all the planes involved
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in trying to sort of see the
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intersection point here in two
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dimensions it's pretty easy because we
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have a lot of
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garetty and we can graph things we said
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ha there's there's a unique point of
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intersection we can see that
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geometrically now we could also do that
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in in here but it would be a little bit
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you know more involved but happily we
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don't really have to draw accurate
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diagram we're not determining the
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solutions geometrically we're
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determining them algebraically
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okay we're solving a system so let's see
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if we can do that and then see what we
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get okay so we'll give these equations
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some names let's say that's equation 1
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equation 2 and equation 3 okay so let's
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see what happens ok so let's take
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equation I'm going to replace these
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equations so let's take patient 2 minus
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2 times equations 1 okay I'm going to
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write that over here equation 2 minus 2
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times equation 1 for the purposes of
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removing X 1 leaving X 1 so when we do
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that we're going to get 0 X 1 ok and
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then how many X 2's well minus X 2 minus
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2 times minus X 2 so that's minus X 2
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plus 2 X 2 so for a total of X 2 and
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then when we take equation 2 minus 2
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times equation 1 the X 3 component will
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be minus 2 X 3 and on the right hand
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side we're gonna get 3 minus 2 times 1
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which is 1 ok so that's good we've
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eliminated X 1 from the first two
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equations and now what we're going to do
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is we're going to eliminate X 1 from the
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first and the third equations we're
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going to take equation 3 and just
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subtract equation 1 what I'm doing here
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is I'm taking this value in this value
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and I'm subtracting to get rid of the X
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ones from equations 1 & 3
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okay what happens there uh let's see do
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that okay so we're gonna get 0 X 1 as
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before so equation 3 minus equation 2
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soul together they'll be minus X 2 + 2 X
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3 minus X 3 will be a single X 3 so plus
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X 3 and on the right hand side we'll get
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minus 2 minus 1 so a total of minus 3 so
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now we are could forget about the
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original equations now we have two more
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equations let's call them box a 4 5 now
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these are two equations but only in X 2
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and X 3
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okay so we've reduced the problem from
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three equations in three variables to
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two equations in two variables okay
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we've done that by getting rid of the
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first variable by taking appropriate
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linear combinations so now we're in
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position as we were before now we want
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to reduce by eliminating one more
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variable let's say X 2 how could we
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eliminate X 2 we could take equation 5
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and we could just add equation 4 okay so
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that's gonna be 0 X 2 minus X 3 equals
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some of those ones is minus 2 from which
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we deduce that X 3 equals 2 and then we
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plug that back in sub let's say into
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equation 4 to get X 2 minus 2 x 2 equals
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1 so therefore X 2 equals 1 plus 4 which
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would be 5 okay now we've got X 3 and X
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2 now we can sub both of those back into
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one of the original equations subset
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equation one I'll remind you what that
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was it was X 1 minus X 2 plus X 3 equals
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1 I'll write that down again X 1 minus X
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2 plus X 3 equals 1 so we're
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substituting these values into there
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we get x1 minus 5 plus 2 equals 1 and so
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x1 equals minus 3 all together up for I
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guess x1 equals 4 so the solution is
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unique we get a unique solution unique
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solution which is X 1 X 2 X 3 equals the
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point 4 5 + 2 4 or 5 integrate did we do
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that right
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I mean there's some arithmetic involved
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here let's check let's go back to the
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original equation so 4 or 5 - so if I
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look at the second equation okay if I
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plug in for 5 - I'm going to get 8 minus
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5 that is 3 that works but plug into the
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third equation it's 4 5 2 so 4 minus 10
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plus 4 is indeed - tube so that works as
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well and for the first equation 4 minus
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5 plus 2 that is indeed 1 so yes that
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point that we found really does lie on
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all three planes okay where is this
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point 4 5 what do we say 4 5
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- ok 4 5 did so it'll be a point on this
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diagram
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I'll be over 4 in this in this x1
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direction so maybe around here and then
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about 5 in this direction and 5 now so
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make that over here okay and 2 in the in
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the X 3 direction so up here so let's
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wait to go here and then over about here
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and then up here like this so this is
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sort of the point that we're going to
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get to the point that we're going to get
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to these three planes whatever they look
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like they're going to intersect in one
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point ok so here's one plane here's
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another plane so if I intersect two
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planes ok I'm going to get a line ok the
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intersection of two planes is generally
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line unless
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unless the planes are parallel in which
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case there is no intersection or they're
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parallel and they're coinciding in which
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case well there's a whole plane of
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points so these are the three
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possibilities just two planes if they
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meet like this we get a line of
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intersection if they're like this
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they're parallel there there's zero
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intersection and if they coincide well
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then there's none bounded number of
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solutions namely every point on that
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plane okay so what happens if we have
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three planes so maybe those two okay and
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then there's a third one like that one
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okay what's the common intersection
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gonna be well any two of them intersect
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on the line so just these two there's a
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line intersection and then this plane is
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gonna meet that line in at most one
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point so in this case would be that
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point right there so these three planes
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typically meet in one point and that's
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what's happening in this example that
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I've just shown you but that's not the
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only possibility right there's other
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things that could happen for example
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geometrically we can have two planes
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that meet in a line like this and then
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we can have a third plane just
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judiciously poised like this so that it
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also passes through that common line of
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intersection okay so in that case then
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the common meet of those three planes
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would be a line that's another
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possibility another possibility is that
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these two planes meet in unbounded
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number of points and then this third
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plane also it happens to coincide with
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the first two so that all three planes
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are basically the same well in that case
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the common solutions will be a plane of
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points still an unbounded number but a
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plane of points and then there's one
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more possibility that you get by having
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three parallel planes like this one here
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at this level
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this one looks like that parallel and
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this one like this parallel okay then
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those three planes are all parallel and
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there's no there's no
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intersection another way that no
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intersection can happen is the following
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we could have the first to meet at a
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line like this okay so there's the the
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meet of that of those first two and then
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the third plane just manages to miss
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that line how does it do that by having
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that same direction vector and as one of
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its direction vectors okay
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so there's actually quite a lot of
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possibilities geometrically so I'll
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summarize that by saying that
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geometrically three planes in 3d can ok
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there's various possibilities first of
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all they can meet in a single in a
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single point as in the example or they
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can meet in a line okay maybe you know
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something like this there's one and
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there's another plane and they meet in
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this line right here that's that's the
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line of intersection and then there's a
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third plane make it like this which also
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meets in that red line so that red line
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is the meet of these three planes and
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when the other possibility we said well
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they could also or they could also meet
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in a plane or they could have no common
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have no common intersection so in terms
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of numbers of solution this would be a
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unique solution
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this first one unique solution and then
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these two whether they meet in a line or
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plane that would be unbounded number of
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solutions
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and this other possibility this one here
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that would be no solution so this is
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really representative of what happens
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with higher dimensions if we have more
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variables we have more equations in our
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system still the same kind of phenomena
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is going to happen as is happening here
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in these three dimensional examples that
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I've what I've worked out one and I've
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shown you geometrically some other
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possibilities so typically we're gonna
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have to do some analysis right we're
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gonna have to investigate to see which
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of these various possibilities arise and
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you know and then to describe the
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solutions there's a line of solutions we
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want to describe that line there's a
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plane of solutions we want to describe
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that plane of solutions if there's a
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single point of course we want to obtain
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that point so what we've done today is
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we've outlined a very broad program for
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solving systems of linear equations and
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we're going to develop technology to be
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able to do that systematically so we
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want to transcend some high school
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methods that I've been showing you today
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and and adopt more matrix oriented
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methods that will very systematically
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show us which of these various
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possibilities are happening and actually
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give us a good way of describing all the
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solutions for a given system linear
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equations so I'll see you next time