Cramer's rule, explained geometrically

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<h4 id="title" class="style-scope ytd-metadata-row-renderer">Caption authors (Polish)</h4>

<div id="content" class="style-scope ytd-metadata-row-renderer"><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCLY0sVSIlw5qERRhLSMk0tA">Fundacja Edukacja Przyszłości</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCQYPNNQExri6AG5NmQjmvNg">KhanAcademyPoPolsku</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCDUgc5ul2JYKTq_FAnadw4A">Lech Mankiewicz</a><a class="yt-simple-endpoint style-scope yt-formatted-string" spellcheck="false" href="https://www.youtube.com/channel/UCBmbEBj4eybdApFesQCcc2w">Centrum Fizyki Teoretycznej PAN</a></div>

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SHOW LESS</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In a previous video, I’ve talked about linear systems of equations, and I sort of brushed</div>

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<div class="cue style-scope ytd-transcript-body-renderer">aside the discussion of actually computing solutions to these systems.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And while it’s true that number-crunching is something we typically leave to the computers,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">digging into some of these computational methods is a good litmus test for whether or not you</div>

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<div class="cue style-scope ytd-transcript-body-renderer">actually understand what’s going on, since this is really where the rubber meets the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Here I want to describe the geometry behind a certain method for computing solutions to</div>

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<div class="cue style-scope ytd-transcript-body-renderer">these systems, known as Cramer’s rule.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The relevant background needed here is an understanding of determinants, dot products,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and of linear systems of equations, so be sure to watch the relevant videos on those</div>

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<div class="cue style-scope ytd-transcript-body-renderer">topics if you’re unfamiliar or rusty.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But first!</div>

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<div class="cue style-scope ytd-transcript-body-renderer">I should say up front that Cramer’s rule is not the best way for computing solutions</div>

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<div class="cue style-scope ytd-transcript-body-renderer">to linear systems of equations.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Gaussian elimination, for example, will always be faster.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So why learn it?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Think of this as a sort of cultural excursion; it’s a helpful exercise in deepening your</div>

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<div class="cue style-scope ytd-transcript-body-renderer">knowledge of the theory of these systems.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Wrapping your mind around this concept will help consolidate ideas from linear algebra,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">like the determinant and linear systems, by seeing how they relate to each other.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Also, from a purely artistic standpoint, the ultimate result is just really pretty to think</div>

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<div class="cue style-scope ytd-transcript-body-renderer">about, much more so that Gaussian elimination.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Alright, so the setup here will be some linear system of equations, say with two unknowns,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">x and y, and two equations.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In principle, everything we’re talking about will work systems with a larger number of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">unknowns, and the same number of equations.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But for simplicity, a smaller example is nicer to hold in our heads.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So as I talked about in a previous video, you can think of this setup geometrically</div>

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<div class="cue style-scope ytd-transcript-body-renderer">as a certain known matrix transforming an unknown vector, [x; y], where you know what</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the output is going to be, in this case [-4; -2].</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Remember, the columns of this matrix tell you how the matrix acts as a transform, each</div>

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<div class="cue style-scope ytd-transcript-body-renderer">one telling you where the basis vectors of the input space land.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So this is a sort of puzzle, what input [x; y], is going to give you this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">output [-4; -2]?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Remember, the type of answer you get here can depend on</div>

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<div class="cue style-scope ytd-transcript-body-renderer">whether or not the transformation squishes all of space into a lower dimension.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">That is if it has zero determinant.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In that case, either none of the inputs land on our given output or there are a whole bunch</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of inputs landing on that output.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But for this video we’ll limit our view to the case of a non-zero determinant, meaning</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the output of this transformation still spans the full n-dimensional space it started in;</div>

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<div class="cue style-scope ytd-transcript-body-renderer">every input lands on one and only one output and every output has one and only one input.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">One way to think about our puzzle is that we know the given output vector is some linear</div>

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<div class="cue style-scope ytd-transcript-body-renderer">combination of the columns of the matrix; x(the vector where i-hat lands) + y(the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">vector where j-hat lands), but we wish to compute what exactly x and y are.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">As a first pass, let me show an idea that is wrong, but in the right direction.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The x-coordinate of this mystery input vector is what you get by taking its dot product</div>

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<div class="cue style-scope ytd-transcript-body-renderer">with the first basis vector, [1; 0].</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Likewise, the y-coordinate is what you get by dotting it with the second basis vector,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So maybe you hope that after the transformation, the dot products with the transformed version</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of the mystery vector with the transformed versions of the basis vectors will also be</div>

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<div class="cue style-scope ytd-transcript-body-renderer">these coordinates x and y.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">That’d be fantastic because we know the transformed versions of each of these vectors.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">There’s just one problem with this: it’s not at all true!</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For most linear transformations, the dot product before and after the transformation will be</div>

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<div class="cue style-scope ytd-transcript-body-renderer">very different.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For example, you could have two vectors generally pointing in the same direction, with a positive</div>

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<div class="cue style-scope ytd-transcript-body-renderer">dot product, which get pulled away from each other during the transformation, in such a</div>

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<div class="cue style-scope ytd-transcript-body-renderer">way that they then have a negative dot product.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Likewise, if things start off perpendicular, with dot product zero, like the two basis</div>

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<div class="cue style-scope ytd-transcript-body-renderer">vectors, there’s no guarantee that they will stay perpendicular after the transformation,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">preserving that zero dot product.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In the example we were looking at, dot products certainly aren’t preserved.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">They tend to get bigger since most vectors are getting stretched.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">In fact, transformations which do preserve dot products are special enough to have their</div>

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<div class="cue style-scope ytd-transcript-body-renderer">own name: Orthonormal transformations.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">These are the ones which leave all the basis vectors perpendicular to each other with unit</div>

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<div class="cue style-scope ytd-transcript-body-renderer">lengths.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">You often think of these as rotation matrices.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The correspond to rigid motion, with no stretching, squishing or morphing.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Solving a linear system with an orthonormal matrix is very easy: Since dot products are</div>

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<div class="cue style-scope ytd-transcript-body-renderer">preserved, taking the dot product between the output vector and all the columns of your</div>

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<div class="cue style-scope ytd-transcript-body-renderer">matrix will be the same as taking the dot products between the input vector and all</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the basis vectors, which is the same as finding the coordinates of the input vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So, in that very special case, x would be the dot product of the first column with the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">output vector, and y would be the dot product of the second column with the output vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Now, even though this idea breaks down for most linear systems, it points us in the direction</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of something to look for: Is there an alternate geometric understanding for the coordinates</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of our input vector which remains unchanged after the transformation?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If your mind has been mulling over determinants, you might think of this clever idea: Take</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the parallelogram defined by the first basis vector, i-hat, and the mystery input vector</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The area of this parallelogram is its base, 1, times the height perpendicular to that</div>

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<div class="cue style-scope ytd-transcript-body-renderer">base, which is the y-coordinate of our input vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So, the area of this parallelogram is sort of a screwy roundabout way to describe the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">vector’s y-coordinate; it’s a wacky way to talk about coordinates, but run with me.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Actually, to be more accurate, you should think of the signed area of this parallelogram,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">in the sense described by the determinant video.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">That way, a vector with negative y-coordinate would correspond to a negative area for this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">parallelogram.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Symmetrically, if you look at the parallelogram spanned by the vector</div>

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<div class="cue style-scope ytd-transcript-body-renderer">and the second basis vector, j-hat, its area will be the x-coordinate of the vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Again, it’s a strange way to represent the x-coordinate, but you’ll see what it buys</div>

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<div class="cue style-scope ytd-transcript-body-renderer">us in a moment.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Here’s what this would look like in three-dimensions: Ordinarily the way you might think of one</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of a vector’s coordinate, say its z-coordinate, would be to take its dot product with the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">third standard basis vector, k-hat.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But instead, consider the parallelepiped it creates with the other two basis vectors,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If you think of the square with area 1 spanned by i-hat and j-hat as the base of this guy,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">its volume is the same its height, which is the third coordinate of our vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Likewise, the wacky way to think about any other coordinate of this vector is to form</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the parallelepiped between this vector an all the basis vectors other than the one you’re</div>

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<div class="cue style-scope ytd-transcript-body-renderer">looking for, and get its volume.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Or, rather, we should talk about the signed volume of these parallelepipeds, in the sense</div>

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<div class="cue style-scope ytd-transcript-body-renderer">described in the determinant video, where the order in which you list the three vectors</div>

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<div class="cue style-scope ytd-transcript-body-renderer">matters and you’re using the right-hand rule.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">That way negative coordinates still make sense.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Okay, so why think of coordinates as areas and volumes like this?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">As you apply some matrix transformation, the areas of the parallelograms don’t stay the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">same, they may get scaled up or down.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">But(!), and this is a key idea of determinants, all these areas get scaled by the same amount.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Namely, the determinant of our transformation matrix.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">For example, if you look the parallelogram spanned by the vector where your first basis</div>

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<div class="cue style-scope ytd-transcript-body-renderer">vector lands, which is the first column of the matrix, and the transformed version of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Well, this is the transformed version of that parallelogram we were looking at earlier,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">whose area was the y-coordinate of the mystery input vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So its area will be the determinant of the transformation multiplied by that value.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So, the y-coordinate of our mystery input vector is the area of this parallelogram,</div>

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<div class="cue style-scope ytd-transcript-body-renderer">spanned by the first column of the matrix and the output vector, divided by the determinant</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of the full transformation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Well, we know the coordinates for where the mystery input vector lands, that’s the whole</div>

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<div class="cue style-scope ytd-transcript-body-renderer">point of a linear system of equations.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So, create a matrix whose first column is the same as that of our matrix, and whose</div>

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<div class="cue style-scope ytd-transcript-body-renderer">second column is the output vector, and take its determinant.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So look at that; just using data from the output of the transformation, namely the columns</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of the matrix and the coordinates of our output vector, we can recover the y-coordinate of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">our mystery input vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Likewise, the same idea can get you the x-coordinate.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Look at that parallelogram we defined early which encodes the x-coordinate of the mystery</div>

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<div class="cue style-scope ytd-transcript-body-renderer">input vector, spanned by the input vector and j-hat.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The transformed version of this guy is spanned by the output vector and the second column</div>

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<div class="cue style-scope ytd-transcript-body-renderer">of the matrix, and its area will have been multiplied by the determinant of the matrix.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So the x-coordinate of our mystery input vector is this area divided by the determinant of</div>

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<div class="cue style-scope ytd-transcript-body-renderer">the transformation.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Symmetric to what we did before, you can compute the area of that output parallelogram by creating</div>

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<div class="cue style-scope ytd-transcript-body-renderer">a new matrix whose first column is the output vector, and whose second column is the same</div>

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<div class="cue style-scope ytd-transcript-body-renderer">as the original matrix.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">So again, just using data from the output space, the numbers we see in our original</div>

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<div class="cue style-scope ytd-transcript-body-renderer">linear system, we can recover the x-coordinate of our mystery input vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">This formula for finding the solutions to a linear system of equations is known as Cramer’s</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Here, just to sanity check ourselves, plug in the numbers here.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The determinant of that top altered matrix is 4+2, which is 6, and the bottom determinant</div>

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<div class="cue style-scope ytd-transcript-body-renderer">is 2, so the x-coordinate should be 3.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">And indeed, looking back at that input vector we started with, it’s x-coordinate is 3.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Likewise, Cramer’s rule suggests the y-coordinate should be 4/2, or 2, and that is indeed the</div>

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<div class="cue style-scope ytd-transcript-body-renderer">y-coordinate of the input vector we started with here.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">The case with three dimensions is similar, and I highly recommend you pause to think</div>

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<div class="cue style-scope ytd-transcript-body-renderer">it through yourself.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Here, I’ll give you a little momentum.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">We have this known transformation, given by a 3x3 matrix, and a known output vector, given</div>

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<div class="cue style-scope ytd-transcript-body-renderer">by the right side of our linear system, and we want to know what input vector lands on</div>

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<div class="cue style-scope ytd-transcript-body-renderer">this output vector.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">If you think of, say, the z-coordinate of the input vector as the volume of this parallelepiped</div>

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<div class="cue style-scope ytd-transcript-body-renderer">spanned by i-hat, j-hat, and the mystery input vector, what happens to the volume of this</div>

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<div class="cue style-scope ytd-transcript-body-renderer">parallelepiped after the transformation?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">How can you compute that new volume?</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Really, pause and take a moment to think through the details of generalizing this to higher</div>

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<div class="cue style-scope ytd-transcript-body-renderer">dimensions; finding an expression for each coordinate of the solution to larger linear</div>

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<div class="cue style-scope ytd-transcript-body-renderer">systems.</div>

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<div class="cue style-scope ytd-transcript-body-renderer">Thinking through more general cases and convincing yourself that it works is where all the learning</div>

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<div class="cue style-scope ytd-transcript-body-renderer">will happen, much more so than listening to some dude on YouTube walk through the reasoning</div>

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<div class="cue style-scope ytd-transcript-body-renderer">again.</div>

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<div id="label-text" class="style-scope yt-dropdown-menu">English</div>

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Cramer's rule, explained geometrically

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