The k-Poly Algebra and truncations

We introduce finite algebraic approximations to the algebra of polynumbers called k-polys, where k is a natural number. The key notion here is that of an algebra: which is a linear or vector space with an additional (associative) multiplication that distributes with the linear structure of the underlying vector space -- over some field.

Examples of algebras include the n x n matrices, the complex numbers which can be considered in both the blue (Euclidean), red and green flavours, the latter two coming from relativistic geometry.

Inverses play a big role in the discussion of algebras.

Truncation operators are important in connecting the algebras of k-polys with the larger algebras of polynumbers, or polyseries. We get an important connection between the world of polynumbers or polyseries and the world of matrix or maxel arithmetic.

This video is part of the Algebraic Calculus Two course, which is currently under development.

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[Music]

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Hello everyone i'm Norman Wildberger welcome to algebraic calculus 2. today's video is called

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the k-poly algebras and truncations we're going to move a little bit into abstract algebra in

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particular we're going to study this concept of an algebra and i'm going to introduce an important

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new family of algebras which we're going to denote by the term k poly where k is a natural number

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these things are in some sense sort of finite and bounded versions of the more familiar poly numbers

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whose type is poly and i remind you that one of our main aims is to try to get at these

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ongoing polynomials or poly numbers which is a type of poly series okay so here is a regular

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poly number 4 0 3 minus 11. it's essentially just a list of numbers but we know that this can be

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translated to the more familiar polynomial form 4 plus 3 alpha squared minus 11 alpha cubed here

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is a more dubious kind of poly series with a pattern that is kind of ongoing in some sense

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1 4 9 16 and then there are three dots so we're still wrestling as to you know how do we how do we

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bring this kind of object into being precisely in particular we really need to focus on the

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nature of these three dots what does this actually mean computationally or to put it

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another way you know how is a computer scientist supposed to understand what's going on here okay

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but today we're going to go kind of in the opposite direction and become even

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more concrete even more finite even more bounded with these k-polys so here is what a k-poly is

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it's a list of numbers for us rational numbers typically in this case two minus one zero five

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but notice that in contrast with the sign for the poly which is this horizontal bar

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on top and then a vertical one on the side this one here has a bar on the top and on the bottom

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okay so it's it's constrained in in extent and you know so the elements are two minus one zero five

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and this is going to be a three poly because this is really degree three here and we're

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going to introduce a similar kind of notation like this so we're going to call this 2 minus

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alpha 3 plus 5 alpha 3 cubed so we're kind of kind of introducing a variant of alpha that will apply

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in the 3 poly situation to capture essentially the same idea but a little bit differently

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and sometimes when we are wanting to be a little bit faster and avoid a subscript we'll just use

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a generic epsilon okay so we'll also write this as 2 minus epsilon plus 0 epsilon squared plus

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5 epsilon cubed now all of these things are algebras okay and so that's going to

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be a key point today is to introduce you to this abstract algebraic object of an algebra

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okay so what is the definition of an algebra so for us at algebra with identity

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in the theory of algebras there's also algebras without identity but they're not as sort of

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central as these ones so we're not always going to say it but all the algebras that we're going

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to be working with always have an identity so what is that well it's first of all a vector or

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linear space over the rationals maybe it's called a so it's a vector space it's got elements that

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can be added and scalar multiplied and the usual vector space properties hold but in addition there

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is a multiplication say denoted a times b and this multiplication has a number of properties

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first of all associativity is required a times b times c equals a times b times c secondly

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it should be distributive with respect to the addition in the linear space so a times b plus c

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should be a times b plus a times c and we use the usual convention that we do these multiplications

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first before we do the addition and similarly on the other side a plus b times c is a times

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c plus b times c and there's another kind of distributive kind of law that concerns itself

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with the operation of scalar multiplication in our vector space so if we have two elements a and

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b we multiply a by say number lambda so lambda a times b well that's the same as what you get when

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you multiply a by lambda b and that's the same as what you get when you multiply a times b by

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lambda so you can move these scalars around in the sort of obvious ways and then there's the crucial

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property that there's an identity and that means that this element one it's a distinguished

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element in the algebra has the property that 1 times a equals a times 1 equals a for any a in

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the algebra okay so that's an algebra okay so there's a quintessential example of an algebra

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it's the central example in mathematics for an algebra and that's the algebra of n by n matrices

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again over whatever field we're working with say over the rationals

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so that is an algebra because you can add matrices you can scale or multiply them and crucially you

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can multiply them the various properties are holding and the identity element is the n by n

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matrix which has ones down the diagonal and zeros elsewhere and this crucially is a non-commutative

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algebra okay so the commutativity is not a required property this is a non-commutative

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algebra and plays a big role it'll actually play a big role for us in the algebraic calculus 2 course

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now some other ones that are also pretty familiar here are the complex numbers

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which i'm going to call them c sub b b standing for blue okay so one way of defining the blue

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complex numbers or the familiar ones it's vectors x y the usual additive structure of row vectors

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and then here is the multiplication x1 y1 times x2 y2 equals x1 x2 minus y1 y2 and x1 y2 plus x2 y1

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now this multiplication can be maybe elucidated by realizing that you can identify this vector

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x y with this 2 by 2 matrix this embeds the blue complex numbers inside the algebra of

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dihedrons which is the algebra of 2x2 matrices special case of the of these ones here so x y

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is associated to the matrix x y minus yx and and then the multiplication of these ones here defined

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up here corresponds to the multiplication for corresponding two by two matrices and importantly

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this algebra is commutative so it obeys this additional law that say z times w equals w times z

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okay so there's two important variants to the blue complex numbers namely the red complex numbers

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and the green complex numbers and they are also algebras so the red complex numbers consisting of

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vectors x y with the usual additive structure and here's the product uh in the red algebras

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situation so x1 y1 times x2 y2 is x1 x2 plus y1 y2 and also x1 y2 plus x2 y1 and we can realize these

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these red complex numbers or represent them by associating to the vector xy the

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by 2 matrix x y y x so then the the multiplication that's being recorded here is just a reflection

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of the matrix multiplication of those kinds of uh 2x2 matrices or dihedrons and then there's

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an another example which is also sort of a relativistic kind of variant of this one the

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green complex numbers also consisting of vectors x y but now the multiplication is very simple

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just a pointwise multiplication x1 y1 times x2 y2 is x1 times x2 y1 times y2 and here the

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association of the vector to a matrix is very simple we just put x and y down the diagonal

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so these are all all three of them are algebras now students of mathematics will be of course very

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familiar with the blue complex numbers because it's a field so what is a field well a field is

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like an algebra but with a more more additional properties and that has to do with inverses okay

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so what makes an algebra a field is the property that pretty well everything except for the zero

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vector has an inverse so let's define invertible elements if a is an algebra with identity one

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then an element a in a is invertible precisely when a times b equals b times

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a equals 1 for some b and a and here we are adopting the more usual convention

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that we don't always write the multiplication with a cross we just write a b for a times b

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now in the examples in the matrix algebra mat n a matrix a is invertible in other words satisfies

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this property precisely when it's determinant is non-zero that's a key result of linear algebra

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in the complex numbers the story is the following in the blue complex numbers a

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vector x y is invertible precisely when x squared plus y squared is not equal to zero

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in fact that's also a determinant that's the determinant of the matrix x y minus y x so it's

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really a kind of a special case of this thing here and similarly in the red complex numbers

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x y is invertible precisely when x squared minus y squared is not equal to zero that's

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the determinant of the matrix x y y x and finally in the green complex numbers x y is invertible

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precisely what x times y is not equal to 0 that's the diagonal of the product of the diagonals which

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is the determinant in that case okay so what makes the blue complex numbers particularly

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sort of familiar to undergraduates and important is that this algebra is a field

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and what that means is that in fact every non-zero element in the algebra is actually invertible

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so this condition applies to all the elements in the space except for the zero zero zero vector

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that's not the case with these ones here you can you can check you can find non-zero elements that

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are non-invertible but i don't want to actually go in that direction of a field because it's

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not the the concept of a field that i'm really interested in here today is the

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concept of an algebra so while the invertibility will play a very important role we do not insist

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that you know everything except for the origin is is invertible

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we just have an algebra then asking you know what are the invertible elements is a very

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you know natural step so now let me introduce the main the main objects of today's lecture

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which are these algebras called k-poly algebras okay and there's one for every natural number k

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and they are very closely connected to the algebra of polynomials that we've worked with a lot in

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the algebraic calculus one course and also in the algebraic calculus ii course but they are

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different okay so what's the definition of the k poly algebra so choose a k a natural number

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including zero so zero one two three whatever so then here is what a an element in this k-poly

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algebra looks like it is a list a vertical list of numbers rational numbers a 0 a 1 up to a k okay

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enclosed in this kind of bracket kind of border a line on top a vertical line beside a line on the

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bottom so that's what a k poly is in this case here actually uh you know it's a k poly because

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we're going down to uh to k here so all together there's k plus one elements in this k poly

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now here's another k poly b with entries b0 b1 down to bk so the first order of business is to

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just say what is the linear structure and that's the obvious thing we're just thinking of these as

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column vectors so the addition of a and b is just point wise and the scalar multiplication is also

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just point wise so that's very familiar and now we come to the multiplication now this multiplication

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will look familiar to you but i want to emphasize that this is really a new thing even though it is

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very very close to the arithmetic for poly numbers but it is essentially different

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so what is the product of this k-poly with this k-poly a times b what is it

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well it's another k-poly okay it's another k-poly and the the zero element is a zero times b zero

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the one element is a zero times b one plus a one times b zero the tooth element is a zero times

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b two plus a one times b one plus a two times b zero hey you're saying this is just the way

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we multiply poly numbers you're absolutely right it's exactly the same but we go down to level k

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so the entry at level k here will be a0 times bk plus a1 times bk minus 1 all the way down to

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a say a k minus 1 times b1 plus ak times b0 so the sum of these indices is always k in this

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last in this case row so now what makes this different from the algebra of poly numbers

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the difference is that we stop right here okay we stop and we we have this

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border okay so there's a border up on top side border and then there's this border underneath

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and that's finished now it's it's it's done now if you were going to multiply these two things as

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poly numbers you would not end up here you would have to keep going because the next one would be

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sort of total uh sum of degrees is k plus 1. so you could get a 1 times b k plus plus a a2

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times bk minus 2 etc all the way up to plus a k b1 but that term is not involved here

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so now it's a natural question well is this really an algebra

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so this is a great exercise to to stare at this and to convince yourself that it is an algebra

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and at some level it's it's kind of obvious because if you think about the way you multiply

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two poly numbers right so imagine having a poly number here and a poly number here and we've

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chosen sort of a level k what we're doing is we're just ignoring everything below that level okay

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and we know that when we multiply poly numbers you know say like this level here the two level

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the two level of the product only depends on the uh poly number ingredients up to level two so we

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wanna know what the level two guy in the product is we only have to look at the entries a0 a1 a2

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from this one and b0 b1 b2 so we just disregard everything else and it's this fact that makes

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it reasonably easy to check that this really is an algebra i also ought to say right at the beginning

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here that these algebras are commutative that's an important consequence of the

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the law for multiplication that i just gave you if you look at it it's symmetric if you

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interchange the a and the b k paulies then each entry is still the same you're just sort of

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reading it instead of going reading from left to right you read it from right to left

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so let's do some arithmetic say with two polys okay so a two poly has three entries

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so level zero level one level two so here are two two pulleys two one minus three and one zero five

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so the sum is of course just the point y sum 1 2 and multiplying it by a scalar is of course

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obvious if you wanted to multiply this thing by a scalar 4 you'd get 12 4 8. okay what about the

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product of these two so 2 1 minus 3 times 1 0 5 okay so here's the product the zeroth entry is

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2 times 1. the one entry is 2 times 0 plus 1 times 1. the two entry is two times five

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plus one times zero plus minus three times one and that amounts to two one ten minus three is seven

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note that this is different from the product of these poly numbers if you rubbed off that bottom

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bar okay and then they'd be poly numbers then you would have to add some more terms

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okay so the the next term would be um 1 times 5 plus minus 3 times 0 that would be 5. so there'd

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be an extra 5. and then there'd be one past that minus 3 times 5 there'd be a minus 15 as well

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okay but they are not being involved this two poly times this two poly is this two poly

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okay it's pretty easy to check that the identity is what you think it should be

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it's the poly one zero zero and here is a very interesting exercise

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okay very interesting exercise i really want you to do this show that a this two poly

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say a0 a1 a2 is invertible precisely when a0 the top entry is non-zero

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okay so you can obviously then generalize this to a k poly instead of just a two poly and a more

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subtle thing that i want you to start thinking about is okay this thing is invertible what then

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is its inverse okay that's going to be a really important problem for the algebraic calculus

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so let's now talk about the standard basis of the k poly algebra there's a standard basis

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and it all rests upon this fundamental k poly the one that looks like 0 1 0 0 all the way down to

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the bottom 0. so this is just like alpha playing the same role as alpha does in the world of poly

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numbers but i remind you that for alpha and in the world of poly numbers we were not fussed about

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these following zeros we allowed ourselves the flexibility of you know being able to add zeros

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at the bottom of a poly number or remove them okay it's not changing the poly number here with these

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k polys we're going to be much more fussy about that we're going to insist that there's exactly k

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plus 1 entries in this list so we are not allowed to rub off any of these zeros they

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are an integral part of these objects okay so they they behave a little bit different from polys so

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since this is not the same as alpha even though it looks pretty close to it let's give it a related

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name let's call it alpha sub k okay because it's a k poly there's altogether k plus 1 elements here

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so that's alpha sub k then alpha sub k squared will be 0 0 1 and then a bunch of zeros

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and carry on alpha sub k to the k the last one will be all zeros with just a one at the bottom

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and then to sort of to be consistent well what about the one the case when there's just a one

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up at the top well let's call that alpha sub k to the zero so then we have a basis from k to the 0

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alpha sub k to the 1 alpha sub k squared all the way up to alpha sub k to the k

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all together k plus 1 elements in this basis so then for example here is a typical k poly say a

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equals a0 a1 down to a k we can then write that as a linear combination of these basis vectors

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start with this one here so it's a 0 times alpha to the k 0. that one there plus a 1 times this

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vector all the way up to a k times this vector so this just sort of corresponds to the obvious

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usual familiar standard basis for column vectors

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but now these have sort of interpretations as powers of this alpha sub k so we could rewrite

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this thing as a zero alpha k to the zero plus a one alpha k to the one or just alpha sub k

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plus a two alpha sub k squared all the way up to a sub k times alpha sub k to the k

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now we are going to have to observe one more thing about this alpha sub k this element here

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that if we raise it to the k plus first power then we get zero okay this is the zero vector

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so why is that so here is here's alpha k and here's alpha

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k to the k what happens if we multiply those well what you do is you um

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you know you're you're going to calculate these these products but

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you're never going to get there's only a single one in this case there's only a one in this case

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and none of these products that that occur in in the expansion uh will have the product of this

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one times this one so every single entry of the product of this one with this one will be zero

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so this is playing this is a very different property that alpha sub k has from the usual alpha

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now we will sometimes when we're we're knowingly working in k poly we might not want to write

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these subscripts all the time because it gets kind of tedious to keep writing alpha sub k

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so i'm going to introduce a shortcut for that

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okay but the shortcut is not alpha because then there's possibility of confusion with the

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polynomial story here's what the shortcut is going to be we're also going to just call this epsilon

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when you see epsilon maybe you think about an infinitesimal

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and if you've watched my famous math problems lecture on how to set up infinitesimals

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you will probably tweak you know why we're doing this that this element here is a potential

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infinitesimal for us in our in our differential calculus as we will see

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okay and now let's introduce truncations but we're going to have a little bit of a different spin on

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truncations than we did last time we talked about this so we've talked about truncations t sub k and

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the idea there was that truncation is you know you take a polynomial and you just truncate it

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terms up to a certain degree and then no more so we're going to be doing something very similar

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but the the shift is is is important okay we're going to now think of this truncation operator

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t sub k as going from the algebra of poly numbers or poly to the algebra of

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k polys and it's defined as follows so t sub k here is a poly number a0 a1 down to some a sub l

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now l could be bigger than k it could be smaller than k it could be equal to k it doesn't matter

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what we're going to do is we're going to truncate this thing we're going to take the top k plus 1

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elements here so a0 a1 down to a sub k and then we're going to stop so we're going to put a bar

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underneath to ensure that we get a k poly and not a poly so i must say that we potentially

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may need to add zeros because we have to go all the way down the bottom so in other words let me

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illustrate with this example here so uh here let's work in the world of three polys okay so three

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pulleys have four entries zero one two three four so here's a poly number one fourteen minus two six

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seventeen let's take t sub three of that what we do is we we strip off the zeroth one the one

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one the two one and the three one and disregard everything else and just put a bar to close things

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how about t sub three of the poly number five minus seven so remind you in the world of poly

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numbers 5 minus 7 is the same as 5 minus 7 0. it's the same as 5 minus 7 0 0. so we're going to use

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that intuition to just copy this thing but copy the first k plus 1 entries in this case 5 minus 7

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0 0 so 4 entries because we're starting at 0 and going up to 3. so it means we have to sort of add

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two zeros so that we have the right number of entries here so we really have a three poly

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okay so here's the truncation theorem which justifies our our usage of this truncation

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operator it's that if p and q are polys and k is a natural number then the truncation t sub k

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of the product p times q is the product of the truncation of p times the truncation of q and

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another way of saying this is that t sub k from poly to k poly is actually an algebra homomorphism

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so this is cleaner than the last time i talked about truncations where we were

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just working in the context of polynomials and there we had to truncate again on the

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right hand side to make things work out going into the world or the algebra of the k-polys

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means that we have a cleaner sort of algebraic structure so let me illustrate here so here's

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let's say t sub 2. here's two poly numbers two minus one one zero three minus four the product

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of these two things is i guess two minus one six minus eleven four so let's take t sub two of this

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what does that mean well we just take the 0th element the 1th element and the 2th element

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and stop there to get this 2 poly 2 minus 1 6. and now we want to convince ourselves

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that this is the same as t sub k of p times t sub k of q so what's t sub k of p

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the truncation of of this one 2 minus 1 that'll be 2 minus 1 0 we're taking t sub 2 so we end up

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with a 2 poly the truncation of this q is one zero three and so now these are in the world of

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k polys let's just check that we get this so two times one is two and two times zero plus

28:13

minus one times one is minus 1 and 2 times 3 plus minus 1 times 0 plus 0 times 1 is 6 and that's all

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there is so this is indeed correct this really is t sub 2 of 2 minus 1. this really is t sub 2 of

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1 0 3 minus 4 and that illustrates this algebra homomorphism property

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so we've done something which in some sense it's very elementary and and familiar but it's

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important to first of all appreciate this the importance of this algebra structure

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okay and that we're aligning things so that sort of algebraically uh things are working nicely

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with respect to these truncations it's going to be really helpful for us as we move further on

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into the differential calculus in the algebraic calculus 2 course

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so there's much to learn i'm Norman Wildberger thanks for listening. [Music]