# Linear Algebra Blog Post 2: Matrix Transformations

Matrices play a key role in Linear Algebra. An elementary study of these objects can yeild the interpertation that they represent a set of vectors. While this is valid view of matrices, it is not quite as interesting as another perspective of matrices; viewing them as *operators* of vectors.
To understand this view, we must first look at what we mean by *transformations*.

A **Transformation** or **mapping** from $\mathbb{R^n}$ to $\mathbb{R^m}$ is a rule that assigns to each vector $\vec v$ in $\mathbb{R^n}$ a vector $\vec T(x)$ in $\mathbb{R^m}$.
The set $\mathbb{R^n}$ is know as the **domain** of our transformation, while $\mathbb{R^m}$ is known as the **codomain**.

Visually, we can picture the following diagram


Using the diagram, we see that $\mathbb{R^U}$ corresponds to our domain, while $\mathbb{R^V}$ corresponds to our codomian.

Now suppose that $A$ is $m \times n$ matrix and $\vec x$ is a vector in $\mathbb{R^n}$. Consider the **matrix-vector product** $A \vec x$:

$$ A \vec x= \begin{bmatrix} \vec a_1 & \vec a_2 & ... & \vec a_n \end{bmatrix}
\begin{bmatrix} x_1 \\ x_2 \\ . \\. \\. \\ x_n \end{bmatrix}.$$

By the definition of the product of a matrix and a vector, we see that $A \vec x$ will be a vector. Moreover, this vector $A \vec x$ will have entries that are linear combinations of the colums of matrix $A$, with the entries of vector $\vec x$ as weights. In other words, the vector $A \vec x$ can be described by the linear combination

$$A \vec x = x_1 \vec a_1 + x_2 \vec a_2 + \cdots + x_n \vec a_n.$$

Consequently, we see that Matrix $A$ *transforms* vector $\vec x$ in $\mathbb{R^n}$ to vector $A \vec x$. To be more precise, the vector $A \vec x$ is actually in $\mathbb{R^m}$. These two facts lead us to conclude that the product of a matrix $A$ by vector $\vec x$ is actually a *transformation*.
The observation above leads us to a define what is ment by a **Matrix Transformation**.

Let $A$ be an $m \times n$ and $\vec x$ a vector in $\mathbb{R^n}$.
A **Matrix Transformation** is a mapping that associates a vector $\vec x$ in $\mathbb{R^n}$ to a vector $A \vec x$ in $\mathbb{R^m}$.

But what is so special about matrix transformations? As it turns out, a variety of vector transformations can actually be represented as a *Matrix Transformation*. We demonstrate this below.

**Example 1**

Let $T: \mathbb{R^2} \longrightarrow \mathbb{R^2}$ be the transformation defined by

$$\vec T(x) = 3 \begin{bmatrix} \ x_1 \\ x_2 \end{bmatrix}.$$

Find a matrix $A$ representation for the transformation $T$.

We observe that by picking an aribitary vector $\vec x$ in our domain $\mathbb{R^2}$, we can determine *what the transformation is doing to the individual entries of our vectors*. Thus, we see that

$$\vec T(x) = 3 \begin{bmatrix} \ x_1 \\ x_2 \end{bmatrix}.$$
$$ \vec T(x) = \begin{bmatrix} \ 3x_1 \\ 3x_2 \end{bmatrix}$$.
$$ \vec T(x) = \begin{bmatrix} \ 3x_1 \\ 0 \end{bmatrix} + \begin{bmatrix} \ 0 \\ 3x_2 \end{bmatrix}.$$

As we can see, the bottom most equation shows $\vec T(x)$ as a sum of two individual vectors. Furthermore, each vector is multiplied by a scalar of 3. In other words, $\vec T((x)$ is a *Linear Combination of two vectors*. Thus, using the definition of a vector-matrix product we see that we can write $\vec T(x)$ as:

$$ \vec T(x) = \begin{bmatrix} \ 3x_1 \\ 0 \end{bmatrix} + \begin{bmatrix} \ 0 \\ 3x_2 \end{bmatrix}.$$

$$ \vec T(x) = \begin{bmatrix} \ 3 & 0 \\ 0 & 3 \\ \end{bmatrix} \begin{bmatrix} \ x_1 \\ x_2 \end{bmatrix}$$.

Hence, we see that $$A = \begin{bmatrix} \ 3 & 0 \\ 0 & 3 \\ \end{bmatrix}.$$

**Exam Question**

Find a matrix $A$ for the transformation

$$\vec T(x) = \begin{bmatrix} \ 2x_1 - 3x_2 \\ -x_2 - x_1 \end{bmatrix}.$$

$$\text{Citations}$$

Lay, D. C. (2006). Linear Algebra and its Applications. Reading: Pearson AddisonWesley Publishing Company
\
Margalit, D., & Rabinoff, J. (n.d.). Interactive Linear Algebra. Retrieved from https://textbooks.math.gatech.edu/ila/matrix-transformations.html

Section LT Linear Transformations. (2012, October 13). Retrieved from http://linear.ups.edu/jsmath/latest/fcla-jsmath-latestli51.html.

Transformation using matrices (Geometry, Transformations). (n.d.). Retrieved from https://www.mathplanet.com/education/geometry/transformations/transformation-using-matrices.