Examples on Transformation matrix

Example 1: Find the new matrix after transformation using the transformation matrix [Tex]\begin{pmatrix} 2 & -3\\ 1 & 2\\ \end{pmatrix} [/Tex]on the vector A = 5i + 4j.

Solution:

Given transformation matrix is T = [Tex]\begin{pmatrix} 2 & -3\\ 1 & 2\\ \end{pmatrix} [/Tex]

Given vector A = 5i + 4j is written as a column matrix as A = [Tex]\begin{pmatrix} 5\\ 4\\ \end{pmatrix} [/Tex]

Let new matrix after transformation be B, and we have the transformation formula as TA = B

B = TA = [Tex]\begin{pmatrix} 2 & -3\\ 1 & 2\\ \end{pmatrix} [/Tex] x [Tex]\begin{pmatrix} 5\\ 4\\ \end{pmatrix} [/Tex]

B = [Tex]\begin{pmatrix} 2 * 5 + (-3) * 4\\ 1 * 5 + 2 * 4\\ \end{pmatrix} [/Tex]

B = [Tex]\begin{pmatrix} -2\\ 13\\ \end{pmatrix} [/Tex]

B = -2i + 13j

Therefore, the new matrix on transformation is -2i + 13j

Example 2: Find the value of the constant ‘a’ in the transformation matrix [Tex]\begin{pmatrix} 1 & a\\ 0 & 1\\ \end{pmatrix}[/Tex] , which has transformed the vector A = 3i + 2j to another vector B = 7i + 2j.

Solution:

Given vectors are A = 3i + 2j and B = 7i + 2j

These vectors written as column matrices are equal to A = [Tex]\begin{pmatrix} 3\\ 2\\ \end{pmatrix} [/Tex], and B = [Tex]\begin{pmatrix} 7\\ 2\\ \end{pmatrix} [/Tex]

This is shear transformation, where only one component of the matrix is changed.

Given transformation matrix is T = [Tex]\begin{pmatrix} 1 & a\\ 0 & 1\\ \end{pmatrix}[/Tex]

Applying the formula of transformation matrix, TA = B, we have the following calculations:

[Tex]\begin{pmatrix} 1 & a\\ 0 & 1\\ \end{pmatrix}[/Tex] x [Tex]\begin{pmatrix} 3\\ 2\\ \end{pmatrix} [/Tex]= [Tex]\begin{pmatrix} 7\\ 2\\ \end{pmatrix} [/Tex]

[Tex]\begin{pmatrix} 1 × 3 + a × 2\\ 0 × 3 + 1 × 2\\ \end{pmatrix} [/Tex]= [Tex]\begin{pmatrix} 7\\ 2\\ \end{pmatrix}[/Tex]

[Tex]\begin{pmatrix} 3 + 2a\\ 2\\ \end{pmatrix}[/Tex] = [Tex]\begin{pmatrix} 7\\ 2\\ \end{pmatrix}[/Tex]

Comparing the elements of the above two matrices, we can calculate the value of a:

3 + 2a = 7

2a = 7 – 3

2a = 4

a = 4/2 = 2

Therefore, the value of a = 2, and the transformation matrix is [Tex]\begin{pmatrix} 1 & 2\\ 0 & 1\\ \end{pmatrix} [/Tex]

Transformation matrices are the core notions in linear algebra and these can help make advancements in many areas including computer graphics, image processing, and so on. Zero vectors and the corresponding unit vectors provide a compact and generalized manner of applying transformations to vectors or points in a coordinate system.

In this article, we will explore detail about transformation matrices, their basic principles, types various applications and others in detail.