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  • Unfortunately, no one can be told, what the Matrix is. You have to see it for yourself. - Morpheus

  • Surprisingly apt words on the importance of understanding matrix operations visually

  • Hey everyone!

  • If I had to choose just one topic that makes

  • all of the others in linear algebra start to click

  • and which too often goes unlearned the first time a student takes linear algebra,

  • it would be this one: the idea of a linear transformation and its

  • relation to matrices.

  • For this video, I'm just going to focus on what these transformations look like in the

  • case of two dimensions

  • and how they relate to the idea of matrix-vector multiplication.

  • In particular, I want to show you a way to think about matrix-vector multiplication that

  • doesn't rely on memorization.

  • To start, let's just parse this termlinear transformation”.

  • Transformationis essentially a fancy word forfunction”.

  • It's something that takes in inputs and spits out an output for each one.

  • Specifically in the context of linear algebra, we like to think about transformations that

  • take in some vector and spit out another vector.

  • So why use the wordtransformationinstead offunctionif they mean the same thing?

  • Well,

  • it's to be suggestive of a certain way to visualize this input-output relation.

  • You see, a great way to understand functions of vectors is to use movement.

  • If a transformation takes some input vector to some output vector,

  • we imagine that input vector moving over to the output vector.

  • Then to understand the transformation as a whole,

  • we might imagine watching every possible input vector move over to its corresponding output vector.

  • It gets really crowded to think about all of the vectors all at once, each one is an arrow,

  • So, as I mentioned last video, a nice trick is to conceptualize each vector, not as an arrow,

  • but as a single point: the point where its tip sits.

  • That way to think about a transformation taking every possible input vector to some output vector,

  • we watch every point in space moving to some other point.

  • In the case of transformations in two dimensions,

  • to get a better feel for the wholeshapeof the transformation,

  • I like to do this with all of the points on an infinite grid.

  • I also sometimes like to keep a copy of the grid in the background,

  • just to help keep track of where everything ends up relative to where it starts.

  • The effect for various transformations, moving around all of the points in space, is,

  • you've got to admit,

  • beautiful.

  • It gives the feeling of squishing and morphing space itself.

  • As you can imagine, though arbitrary transformations can look pretty complicated,

  • but luckily linear algebra limits itself to a special type of transformation,

  • ones that are easier to understand, calledlineartransformations.

  • Visually speaking, a transformation is linear if it has two properties:

  • all lines must remain lines, without getting curved,

  • and the origin must remain fixed in place.

  • For example, this right here would not be a linear transformation since the lines get all curvy

  • and this one right here, although it keeps the line straight,

  • is not a linear transformation because it moves the origin.

  • This one here fixes the origin and it might look like it keeps line straight,

  • but that's just because I'm only showing the horizontal and vertical grid lines,

  • when you see what it does to a diagonal line, it becomes clear that it's not at all linear

  • since it turns that line all curvy.

  • In general, you should think of linear transformations as keeping grid lines parallel and evenly spaced.

  • Some linear transformations are simple to think about, like rotations about the origin.

  • Others are a little trickier to describe with words.

  • So how do you think you could describe these transformations numerically?

  • If you were, say, programming some animations to make a video teaching the topic

  • what formula do you give the computer so that if you give it the coordinates of a vector,

  • it can give you the coordinates of where that vector lands?

  • It turns out that you only need to record where the two basis vectors, i-hat and j-hat, each land.

  • and everything else will follow from that.

  • For example, consider the vector v with coordinates (-1,2),

  • meaning that it equals -1 times i-hat + 2 times j-hat.

  • If we play some transformation and follow where all three of these vectors go

  • the property that grid lines remain parallel and evenly spaced has a really important consequence:

  • the place where v lands will be -1 times the vector where i-hat landed

  • plus 2 times the vector where j-hat landed.

  • In other words, it started off as a certain linear combination of i-hat and j-hat

  • and it ends up is that same linear combination of where those two vectors landed.

  • This means you can deduce where v must go based only on where i-hat and j-hat each land.

  • This is why I like keeping a copy of the original grid in the background;

  • for the transformation shown here we can read off that i-hat lands on the coordinates (1,-2).

  • and j-hat lands on the x-axis over at the coordinates (3, 0).

  • This means that the vector represented by (-1) i-hat + 2 times j-hat

  • ends up at (-1) times the vector (1, -2) + 2 times the vector (3, 0).

  • Adding that all together, you can deduce that it has to land on the vector (5, 2).

  • This is a good point to pause and ponder, because it's pretty important.

  • Now, given that I'm actually showing you the full transformation,

  • you could have just looked to see the v has the coordinates (5, 2),

  • but the cool part here is that this gives us a technique to deduce where any vectors land,

  • so long as we have a record of where i-hat and j-hat each land,

  • without needing to watch the transformation itself.

  • Write the vector with more general coordinates x and y,

  • and it will land on x times the vector where i-hat lands (1, -2),

  • plus y times the vector where j-hat lands (3, 0).

  • Carrying out that sum, you see that it lands at (1x+3y, -2x+0y).

  • I give you any vector, and you can tell me where that vector lands using this formula

  • what all of this is saying is that a two dimensional linear transformation

  • is completely described by just four numbers:

  • the two coordinates for where i-hat lands

  • and the two coordinates for where j-hat lands.

  • Isn't that cool?

  • it's common to package these coordinates into a two-by-two grid of numbers,

  • called a two-by-two matrix,

  • where you can interpret the columns as the two special vectors

  • where i-hat and j-hat each land.

  • If you're given a two-by-two matrix describing a linear transformation

  • and some specific vector

  • and you want to know where that linear transformation takes that vector,

  • you can take the coordinates of the vector

  • multiply them by the corresponding columns of the matrix, then add together what you get.

  • This corresponds with the idea of adding the scaled versions of our new basis vectors.

  • Let's see what this looks like in the most general case

  • where your matrix has entries a, b, c, d

  • and remember, this matrix is just a way of packaging the information needed to describe

  • a linear transformation.

  • Always remember to interpret that first column, (a, c),

  • as the place where the first basis vector lands

  • and that second column, (b, d), is the place where the second basis vector lands.

  • When we apply this transformation to some vector (x, y), what do you get?

  • Well,

  • it'll be x times (a, c) plus y times (b, d).

  • Putting this together, you get a vector (ax+by, cx+dy).

  • You can even define this as matrix-vector multiplication

  • when you put the matrix on the left of the vector

  • like it's a function.

  • Then, you could make high schoolers memorize this,

  • without showing them the crucial part that makes it feel intuitive.

  • But,

  • isn't it more fun to think about these columns

  • as the transformed versions of your basis vectors

  • and to think about the results

  • as the appropriate linear combination of those vectors?

  • Let's practice describing a few linear transformations with matrices.

  • For example,

  • if we rotate all of space 90° counterclockwise

  • then i-hat lands on the coordinates (0, 1)

  • and j-hat lands on the coordinates (-1, 0).

  • So the matrix we end up with has columns (0, 1), (-1, 0).

  • To figure out what happens to any vector after 90° rotation,

  • you could just multiply its coordinates by this matrix.

  • Here's a fun transformation with a special name, called a “shear”.

  • In it, i-hat remains fixed

  • so the first column of the matrix is (1, 0),

  • but j-hat moves over to the coordinates (1,1)

  • which become the second column of the matrix.

  • And, at the risk of being redundant here,

  • figuring out how a shear transforms a given vector

  • comes down to multiplying this matrix by that vector.

  • Let's say we want to go the other way around,

  • starting with the matrix, say with columns (1, 2) and (3, 1),

  • and we want to deduce what its transformation looks like.

  • Pause and take a moment to see if you can imagine it.

  • One way to do this

  • is to first move i-hat to (1, 2).

  • Then, move j-hat to (3, 1).

  • Always moving the rest of space in such a way

  • that keeps grid lines parallel and evenly spaced.

  • If the vectors that i-hat and j-hat land on are linearly dependent

  • which, if you recall from last video,

  • means that one is a scaled version of the other.

  • It means that the linear transformation squishes all of 2D space

  • on to the line where those two vectors sit,

  • also known as the one-dimensional span

  • of those two linearly dependent vectors.

  • To sum up, linear transformations

  • are a way to move around space

  • such that the grid lines remain parallel and evenly spaced

  • and such that the origin remains fixed.

  • Delightfully,

  • these transformations can be described using only a handful of numbers.

  • The coordinates of where each basis vector lands.

  • Matrices give us a language to describe these transformations

  • where the columns represent those coordinates

  • and matrix-vector multiplication is just a way to compute

  • what that transformation does to a given vector.