Transformation Using Matrices

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.

Related Topics

• How to Find Addition and Subtraction of Vectors
• How to Find Scalar Multiplication of Vectors

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 \(\begin{bmatrix}2 & -3 \\1 & 2\end{bmatrix}\).

• \(\color{blue}{\begin{bmatrix} 6 \\12\end{bmatrix}}\)
• \(\color{blue}{\begin{bmatrix} 8 \\6\end{bmatrix}}\)
• \(\color{blue}{\begin{bmatrix} 2 \\-6\end{bmatrix}}\)
• \(\color{blue}{-2i+13j}\)