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})

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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 & -2end{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 & 2end{bmatrix}).

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