Linear transformations and matrices | Chapter 3, Essence of linear algebra

Summary

This video script explores the concept of linear transformations and their relationship with matrices in linear algebra. It simplifies the understanding of matrix-vector multiplication by visualizing transformations as movements of vectors in a two-dimensional space. The script explains that linear transformations maintain grid lines parallel and evenly spaced, with the origin fixed, and can be numerically described using matrices. It emphasizes that matrices represent the outcome of these transformations on basis vectors, allowing us to calculate the new position of any vector through matrix-vector multiplication.

Takeaways

• 🔄 Linear transformations are functions that map vectors to other vectors while preserving the structure of space.
• 🧠 The concept of linear transformations is foundational to understanding linear algebra.
• 📏 A linear transformation must keep lines straight and the origin fixed to be considered linear.
• 📐 Visualizing transformations as movements of vectors helps in understanding their effects.
• 🌐 Linear transformations can be represented by matrices, which describe how basis vectors are transformed.
• 🔢 Matrix-vector multiplication is a numerical method to determine the result of a linear transformation on a vector.
• 📝 Memorizing matrix operations is less effective than understanding their geometric interpretation.
• 🔄 A 2x2 matrix can fully describe a linear transformation in two dimensions by showing where the basis vectors land.
• 🔄 Special types of transformations, like rotations and shears, have characteristic matrices that define their effects.
• 🔑 Understanding matrices as transformations is key to grasping advanced topics in linear algebra, such as eigenvalues and determinants.

Q & A

What is the key concept in linear algebra that the video script emphasizes?

The key concept emphasized in the video script is the idea of a linear transformation and its relation to matrices.

How does the video script suggest understanding functions of vectors?

The video script suggests understanding functions of vectors by visualizing them as movements, where an input vector moves to an output vector.

What is the significance of using an infinite grid to visualize transformations?

Using an infinite grid helps in visualizing how every point in space moves to another point during a transformation, providing a better feel for the whole shape of the transformation.

Why is it important to keep a copy of the original grid in the background during transformations?

Keeping a copy of the original grid helps in tracking where everything ends up relative to where it started, which is crucial for understanding the transformation.

What are the two properties that visually define a linear transformation?

A transformation is linear if it keeps lines straight without curving and if the origin remains fixed in place.

How does the video script describe the numerical representation of linear transformations?

The script describes numerical representation by showing that you only need to record where the two basis vectors, i-hat and j-hat, each land.

What is the importance of basis vectors in linear transformations?

Basis vectors are important because their transformed coordinates determine where any vector lands after the transformation.

How can you describe a two-dimensional linear transformation using a matrix?

A two-dimensional linear transformation can be described using a 2x2 matrix, where the columns represent the coordinates of where i-hat and j-hat land.

What does matrix-vector multiplication represent in the context of linear transformations?

Matrix-vector multiplication represents the process of applying a linear transformation to a vector, resulting in a new vector that is the transformed version of the original.

How does the video script explain the concept of a 90-degree rotation using a matrix?

The script explains a 90-degree rotation by showing that i-hat lands on (0, 1) and j-hat lands on (-1, 0), resulting in a matrix with columns [0, 1] and [-1, 0].

What is a shear transformation and how is it represented by a matrix?

A shear transformation is a linear transformation where one basis vector remains fixed while the other moves. It is represented by a matrix where the first column is [1, 0] if i-hat is fixed, and the second column represents the new position of j-hat.

Outlines

00:00

📐 Understanding Linear Transformations

The first paragraph introduces the concept of linear transformations in the context of linear algebra, focusing on how they relate to matrices and matrix-vector multiplication. It explains that a transformation is a function that maps vectors to other vectors, and the term 'transformation' is used to emphasize the visual aspect of moving from input to output vectors. The idea of representing vectors as points rather than arrows is discussed to simplify the visualization of transformations. The paragraph further clarifies that linear transformations must preserve the straightness of lines and keep the origin fixed. It concludes by explaining how linear transformations can be numerically described by noting the new positions of the basis vectors i-hat and j-hat after the transformation.

05:01

🔢 Describing Linear Transformations with Matrices

The second paragraph delves into how linear transformations can be numerically described using matrices. It explains that knowing the final positions of the basis vectors i-hat and j-hat after a transformation allows us to determine the outcome of any vector transformation without observing the transformation itself. The concept of matrix-vector multiplication is introduced as a method to find the new position of any vector given its original coordinates and the transformation matrix. The paragraph also discusses how a 2x2 matrix can represent a linear transformation, with the columns of the matrix corresponding to the transformed basis vectors. Examples of specific transformations, such as a 90-degree rotation and a shear, are provided to illustrate how matrices represent these transformations.

10:03

🌐 The Power of Matrices in Linear Algebra

The final paragraph emphasizes the importance of understanding matrices as representations of linear transformations in space. It suggests that grasping this concept is crucial for a deeper comprehension of linear algebra topics such as matrix multiplication, determinants, change of basis, and eigenvalues. The paragraph concludes by hinting at the next topic to be covered, which is matrix multiplication, and thanks the viewer for watching.

Keywords

💡Linear Transformation

A linear transformation is a function that maps vectors to vectors while preserving the operations of vector addition and scalar multiplication. In the context of the video, linear transformations are used to describe how space can be 'moved' or 'morphed' while maintaining the parallel and evenly spaced nature of grid lines. An example from the script is a rotation about the origin, which is a linear transformation because it keeps the grid structure intact.

💡Matrix

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In linear algebra, matrices are used to represent linear transformations, with each column representing the image of the basis vectors under the transformation. The video emphasizes that a matrix can be thought of as a compact way to describe a linear transformation, where the columns of the matrix correspond to the new positions of the basis vectors after the transformation.

💡Matrix Vector Multiplication

Matrix vector multiplication is an operation that takes a matrix and a vector and produces another vector. This operation is central to understanding how a linear transformation affects a vector. The video script explains that multiplying a matrix by a vector involves scaling and adding the columns of the matrix (which represent the transformed basis vectors) in accordance with the components of the input vector.

💡Basis Vectors

Basis vectors are vectors that form a basis for a vector space, meaning any vector in that space can be written as a linear combination of these basis vectors. In the video, the basis vectors i-hat and j-hat are used to describe the action of linear transformations in two dimensions. The script illustrates that by knowing where these basis vectors map to under a transformation, one can determine the image of any vector.

💡Transformation

In the video, a transformation refers to any function that maps inputs to outputs, with a focus on those that map vectors to vectors in a way that can be visualized as moving points in space. The video uses the term to describe how linear algebra deals with functions that transform one vector into another, often visualized as moving from one point to another on a grid.

💡Origin

The origin in a coordinate system is the point where the axes intersect, typically at the coordinate (0,0). The video mentions that a linear transformation must keep the origin fixed to be considered linear. This is important because it ensures that the transformation maintains the structure of the space, such as the distances between points.

💡Grid Lines

Grid lines in the context of the video refer to the lines that make up the coordinate grid in two-dimensional space. The video emphasizes that a linear transformation must keep grid lines parallel and evenly spaced, which is a visual way of saying that the transformation must preserve the structure of the space, including the distances between points.

💡Rotation

Rotation is a type of linear transformation that turns a figure around a fixed point without changing its size or shape. In the video, a 90-degree counterclockwise rotation is used as an example of a linear transformation. The script explains that under this transformation, the basis vector i-hat would map to (0,1) and j-hat to (-1,0), which is represented by a specific matrix.

💡Shear

Shear is a type of linear transformation that distorts a shape by sliding each point along a line parallel to a given direction. The video describes a shear transformation where i-hat remains fixed, and j-hat moves to a new position, resulting in a matrix with the first column as (1,0) and the second column as (1,1). This transformation is used to illustrate how non-uniform scaling can be represented by a matrix.

💡Eigenvalues

Eigenvalues are scalar values that describe the amount by which a vector is stretched or shrunk by a linear transformation. Although not explicitly detailed in the script, eigenvalues are mentioned as a topic that becomes easier to understand once one grasps the concept of matrices as transformations. They are a fundamental concept in linear algebra that relates closely to the idea of how transformations affect vectors.

Highlights

The importance of understanding linear transformations and their relation to matrices in linear algebra.

Focusing on two-dimensional linear transformations and their connection to matrix vector multiplication.

A linear transformation is a function that maps vectors to other vectors.

Visualizing transformations as movements of vectors in space.

Conceptualizing vectors as points rather than arrows for easier visualization.

Using an infinite grid to understand the shape of a two-dimensional transformation.

Linear transformations must keep lines straight and the origin fixed.

Describing linear transformations numerically by where the basis vectors land.

The basis vectors i-hat and j-hat determine the outcome of any vector transformation.

Matrix-vector multiplication as a technique to find where a vector lands after a transformation.

A 2x2 matrix can describe a two-dimensional linear transformation with four numbers.

Interpreting matrix columns as the transformed basis vectors.

Matrix-vector multiplication as a way to compute the transformation of a given vector.

Describing the 90-degree rotation transformation with a specific matrix.

Shear transformation and its corresponding matrix representation.

Interpreting a matrix by imagining the transformation it represents.

Linear transformations squishing space onto a line when basis vectors are linearly dependent.

Matrices as a language to describe space transformations.

The connection between matrices and various linear algebra topics like determinants and eigenvalues.

Transcripts

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 term, linear transformation.

Transformation is essentially a fancy word for function.

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 word transformation instead of function if 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 as 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 whole shape of 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, called linear transformations.

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 lines straight,

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

This one here fixes the origin, and it might look like it keeps lines 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 negative 1, 2,

meaning that it equals negative 1 times i-hat plus 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 negative 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 as 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,

negative 2, and j-hat lands on the x-axis over at the coordinates 3, 0.

This means that the vector represented by negative 1 i-hat plus 2 times j-hat

ends up at negative 1 times the vector 1, negative 2 plus 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 that 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, negative 2,

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

Carrying out that sum, you see that it lands at 1x plus 3y, negative 2x plus 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 2x2 grid of numbers called a 2x2 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 2x2 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, AC,

as the place where the first basis vector lands, and that second column, BD,

as the place where the second basis vector lands.

When we apply this transformation to some vector xy, what do you get?

Well, it'll be x times AC plus y times BD.

Putting this together, you get a vector Ax plus By, Cx plus Dy.

You could 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 result 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 degrees counterclockwise,

then i-hat lands on the coordinates 0, 1.

And j-hat lands on the coordinates negative 1, 0.

So the matrix we end up with has columns 0, 1, negative 1, 0.

To figure out what happens to any vector after a 90-degree 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 a 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 gridlines 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 onto 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

gridlines 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.

The important takeaway here is that every time you see a matrix,

you can interpret it as a certain transformation of space.

Once you really digest this idea, you're in a

great position to understand linear algebra deeply.

Almost all of the topics coming up, from matrix multiplication to determinants,

change of basis, eigenvalues, all of these will become easier to understand

once you start thinking about matrices as transformations of space.

Most immediately, in the next video, I'll be talking about

multiplying two matrices together. See you then! Thank you for watching!