Introduction

If a matrix is a machine that moves space, then matrix multiplication is the set of instructions for that machine. In 4.13, we will look at matrices as actual functions, similar to how $f(x)=2x$ is a function, but for vectors instead of single numbers.

Building a Matrix from a Rule

Suppose a linear transformation is given by the rule: 

$(x,y)\to (2x+3y,5x-y)$.

Remember that multiplying a matrix by a vector works like this:

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}ax+by\\ cx+dy\end{array}\right]$.

Notice that the first row produces the first output coordinate, while the second row produces the second output coordinate.

Now, let’s compare this with our transformation rule:

$(2x+3y,5x-y)$.

The first output is 

$2x+3y$,

so the first row of the matrix must be

$\left[\begin{array}{cc}2& 3\end{array}\right]$.

The second output is

$5x-y$,

so the second row must be

$\left[\begin{array}{cc}5& -1\end{array}\right]$.

Therefore, the transformation is represented by 

$A=\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right]$,

since

$A\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}2x+3y\\ 5x-y\end{array}\right]$.

The Unit Vector Trick

A great way to find a matrix is to see what it does to the unit vectors $\hat{i}=\left[\begin{array}{c}1\\ 0\end{array}\right]$ and $\hat{j}=\left[\begin{array}{c}0\\ 1\end{array}\right]$.

The first column of the matrix is where $\hat{i}$ lands, and the second column of the matrix is where $\hat{j}$ lands.

Special Transformations

Rotations

To rotate space counterclockwise by an angle $\theta$, we use the Rotation Matrix:

$R_{\theta }=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$

If you want to rotate $90°$ ($\frac{\pi }{2}$ radians), you plug in the values to get $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$, which was briefly introduced in 4.12.

Dilation and the Determinant

In 4.12, we learned that the determinant $|ad-bc|$ is the area of a parallelogram. In terms of functions, the absolute value of the determinant $|\det (A)|$ tells you the magnitude of dilation.

If $|\det (A)|=5$, any shape you transform will become 5 times as large in area.

If $|\det (A)|=1$, the area stays the same (like in a rotation).

Composition of Matrices

All the way back in Unit 2.7, we introduced Composition of Functions, which is when one entire function is the input to another. In algebra, this looks like $f(g(x))$ or $(f\circ g)(x)$. When working with matrices, we can do $L_1(L_2(\vec{v}))$ or $(L_1\circ L_2)(\vec{v})$. This is called a composition of matrices.

Doing two transformations in a row is the same as multiplying their matrices together:

Matrix of $(L_1\circ L_2)=A_1\cdot A_2$

Note that the composition of linear transformations is itself a linear transformation.

Be careful!

Remember that order matters. $A_1\cdot A_2$ is usually different from $A_2\cdot A_1$.

Inverse Transformations

An inverse transformation $L^{-1}$ is the “Undo” button. If $L$ rotates space $30°$ clockwise, $L^{-1}$ rotates it $30°$ counterclockwise.

Two transformations are inverses if their composition maps every vector back to itself: $L^{-1}(L(\vec{v}))=\vec{v}$

If $L(\vec{v})=A\vec{v}$, then $L^{-1}(\vec{v})=A^{-1}\vec{v}$.

If the determinant is $0$, you can’t “undo” the transformation because space has been “collapsed” into a line or a point.

Practice Problems