# Transformation Using Matrices

A vector can be represented as a column matrix. Here you will learn more about using matrices to transform the plane.

The transformation matrix is a matrix that transforms one vector into another vector. The transformation matrix changes the Cartesian system and plots the vector coordinates to new coordinates.

## A step-by-step guide to transformation using matrices

An ordered pair $$(x,y)$$ can be used to represent a vector, however, a column matrix can also be used:

$$\begin{bmatrix}x \\y \end{bmatrix}$$

Polygons could also be represented in matrix form, we simply place all of the coordinates of the vertices into one matrix. This is called a vertex matrix.

### Transformation Using Matrices – Example 1:

A square has its vertexes in the following coordinates $$(1,1)$$, $$(-1,1)$$, $$(-1,-1)$$ and $$(1,-1)$$. Create a vertex matrix.

To create a vertex matrix, plug each ordered pair into each column of a $$4$$ column matrix:

$$\begin{bmatrix}x_1 & x_2 & x_3 & x_4 \\y_1 & y_2 & y_3 & y_4 \end{bmatrix}$$$$=$$ $$\begin{bmatrix}1 & -1 & -1 & 1 \\1 & 1 & -1 & -1 \end{bmatrix}$$

### Transformation Using Matrices – Example 2:

Create a reflection of the vector in the $$x$$-axis. $$v⃗ = \begin{bmatrix}-1 & 3 \\2 & -2 \end{bmatrix}$$

To create a reflection, we have to multiply it by the correct reflection matrix:

$$\begin{bmatrix}-1 & 0 \\0 & 1 \end{bmatrix}$$

So the vertex matrix of our reflection is:

$$\begin{bmatrix}-1 & 3 \\2 & -2 \end{bmatrix}$$ . $$\begin{bmatrix}-1 & 0 \\0 & 1 \end{bmatrix}$$

$$=\begin{bmatrix}(-1.-1) + (3. \space0) & (-1. \space 0) + (3. \space 1) \\ (2.\space -1) +(-2 .\space 0) & (2. \space 0) (-2.\space 1) \end{bmatrix}$$ $$= \begin{bmatrix}1 & 3 \\-2 & -2\end{bmatrix}$$

## Exercises for Transformation Using Matrices

• If $$v⃗ =(2,4)$$ and $$?⃗ =(6,2)$$. Use matrices to find each of the following vectors:
• $$\color{blue}{3v}$$
• $$\color{blue}{v+u}$$
• $$\color{blue}{u-2v}$$
• Find the new vector formed for the vector $$5i+4j$$, with the help of the transformation matrix $$\begin{bmatrix}2 & -3 \\1 & 2\end{bmatrix}$$.
• $$\color{blue}{\begin{bmatrix} 6 \\12\end{bmatrix}}$$
• $$\color{blue}{\begin{bmatrix} 8 \\6\end{bmatrix}}$$
• $$\color{blue}{\begin{bmatrix} 2 \\-6\end{bmatrix}}$$
• $$\color{blue}{-2i+13j}$$

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