A matrix is more than just a grid of numbers; it is a machine that moves space. When we use a matrix to change a vector into a new vector, we call this a Linear Transformation.

Imagine a coordinate plane made of stretchy rubber. A linear transformation is like grabbing that rubber sheet and stretching, rotating, or flipping it.

What is a Linear Transformation?

In simple terms, a transformation L is a function that takes an input vector $\vec{v}$ and gives you an output vector $L(\vec{v})$. For a transformation to be “linear,” it must follow two strict rules:

The zero vector $\left[\begin{array}{c}0\\ \\ 0\end{array}\right]$ must always stay at $\left[\begin{array}{c}0\\ \\ 0\end{array}\right]$. You can’t slide the whole plane over (that’s called a translation, not a linear transformation).

Grid lines must stay straight and evenly spaced after the transformation.

Every linear transformation from ${\mathbb{R}}^2$ to ${\mathbb{R}}^2$ can be represented by a unique $2\times 2$ matrix A. We write this as:

$L(\vec{v})=A\vec{v}$

Conversely, every $2\times 2$ matrix defines a linear transformation.

Mathematically, a transformation is linear if it preserves vector addition and scalar multiplication:

$L(\mathbf{u}+\mathbf{v})=L(\mathbf{u})+L(\mathbf{v}),L(c\mathbf{v})=cL(\mathbf{v})$

Vectors as Matrices

To use these “matrix machines,” we represent our points as vectors,

A Single Vector

A single point $(x,y)$ in ${\mathbb{R}}^2$ is written as a $2\times 1$ matrix $\vec{v}=\left[\begin{array}{c}x\\ \\ y\end{array}\right]$

Multiple Vectors

If you have a whole shape (like a triangle with three vertices), you can put all those points into one $2\times n$ matrix:

Shape $=\left[\begin{array}{ccc}{x}_1& x_2& x_3\\ & & \\ y_1& y_2& y_3\end{array}\right]$

Transforming a Vector

To find the output vector, you perform matrix multiplication. If $A=\left[\begin{array}{cc}a& b\\ & \\ c& d\end{array}\right]$  and $\vec{v}=\left[\begin{array}{c}x\\ \\ y\end{array}\right]$, the transformation is:

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

Notice how each component of the output is just a constant multiple of the input components added together.

Transforming Multiple Vectors at Once

One aspect of matrices is that they can transform many points simultaneously. If you multiply a $2\times 2$ transformation matrix by a $2\times n$ matrix of input vectors, the result is a $2\times n$ matrix where every column is a new, transformed output vector.

$A\cdot \left[\begin{array}{cccc}{\vec{v}}_1& {\vec{v}}_2& \cdots & {\vec{v}}_n\end{array}\right]=\left[\begin{array}{cccc}L({\vec{v}}_1)& L({\vec{v}}_2)& \cdots & L({\vec{v}}_n)\end{array}\right]$

Common Transformations

Here’s a list of common transformations:

Identity

$\left[\begin{array}{cc}1& 0\\ & \\ 0& 1\end{array}\right]$

Horizontal Dilation

$\left[\begin{array}{cc}2& 0\\ & \\ 0& 1\end{array}\right]$

Reflection across the x-axis

$\left[\begin{array}{cc}1& 0\\ & \\ 0& -1\end{array}\right]$

$90^{\circ }$ rotation

$\left[\begin{array}{cc}0& -1\\ & \\ 1& 0\end{array}\right]$

Practice