Transformation Matrix
Transformation Matrix is a matrix that transforms one vector into another vector by the process of matrix multiplication. The transformation matrix alters the cartesian system and maps the coordinates of the vector to the new coordinates. The transformation matrix T of order m x n on multiplication with a vector A of n components represented as a column matrix transforms it into another matrix representing a new vector A'.
For a twodimensional vector space, the transformation matrix is of order 2 x 2, and for an ndimensional space, the transformation matrix is of order n x n. Let us learn more about the transformation matrix, types of transformation matrices, and application of the transformation matrix.
What Is Transformation Matrix?
Transformation matrix is a matrix that transforms one vector into another vector. The positional vector of a point is changed to another positional vector of a new point, with the help of a transformation matrix. The position vector of a point A = xi + yj, on multiplying with a matrix T = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) is transformed to another vector B = x'i + y'j. Here the vector A = xi + yj is represented as a column matrix A = \(\begin{bmatrix}x\\y\end{bmatrix}\), and the matrix B = x'i + y'j is another column matrix B = \(\begin{bmatrix}x'\\y'\end{bmatrix}\).
The operation of the transformation of vectors using a transformation matrix is as follows.
TA = B
\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)
The transformation matrix can be taken as the transformation of space. Here a 2 x 2 transformation matrix is used for twodimensional space, and a 3 x 3 transformation matrix is used for threedimensional space. The vector in a twodimensional space has two components which is represented as a column vector with two elements, and following the matrix multiplication condition it is transformed using a transformation matrix of order 2 x 2.
Types of Transformation Matrix
The transformation matrix transforms a vector into another vector, which can be understood geometrically in a twodimensional or a threedimensional space. The frequently used transformations are stretching, squeezing, rotation, reflection, and orthogonal projection. Let us learn about some of these transformations in detail.
Stretching
The linear transformation enlarges the distance in the xy plane by a constant value. Here the distance is enlarged or compressed in a particular direction with reference to only one of the axis and the other axis is kept constant. A stretch along the xaxis by keeping the yaxis the same is x' = kx, and y' = y. Here k is a constant numeric value and for k > 1 it is a stretch, k < 1 it is a compression, and for k = 1 it is the same point. The transformation matrix for a stretch along the xaxis is as follows.
T_{x} = \(\begin{bmatrix}k&0\\0&1\end{bmatrix}\)
Also, we can stretch along the yaxis to obtain x' = x, and y' = ky. The transformation matrix for a stretch along the yaxis is as follows.
T_{y} = \(\begin{bmatrix}k&0\\0&1\end{bmatrix}\)
Rotation
The transformation matrix helps to rotate the vector in an anticlockwise direction at an angle θ. The transformation matrix \(\begin{bmatrix}Cosθ&Sinθ\\Sinθ = &Cosθ\end{bmatrix}\) transforms the vector xi + yj to x'i + y'j, which is represented as follows.
\(\begin{bmatrix}Cosθ&Sinθ\\Sinθ&Cosθ\end{bmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)
Shearing
The shearing changes the vector in such a way that the square boxes in the coordinate axes are deformed into parallelograms. The transformation matrix for shearing along the xaxis is \(\begin{bmatrix}1&k\\0&1\end{bmatrix}\) and it transforms the vector A = xi + yj to A' = x'i + y'j such that x' = x + ky, y' = y. This transformation can be understood from the below multiplication of matrices.
\(\begin{bmatrix}1&k\\0&1\end{bmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)
Also, the transformation matrix for shearing along the yaxis is \(\begin{bmatrix}1&0\\k&1\end{bmatrix}\), and it changes changes the vector such that x' = x, and y' = y + kx. The matrix multiplication for this transformation is as follows.
\(\begin{bmatrix}1&0\\k&1\end{bmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)
Application of Transformation Matrix
The transformation matrix has numerous applications in vectors, linear algebra, matrix operations. The following are some of the important applications of the transformation matrix.
 Vectors represented in a two or threedimensional frame are transformed to another vector.
 Linear Combinations of two or more vectors through multiplication are possible through a transformation matrix.
 The linear transformations of matrices can be used to change the matrices into another form.
 Matrix multiplication is the transformation of one matrix into another matrix.
 Determinants can be solved using the concepts of the transformation matrix.
 Inverse Space also use matrix transformations.
 Dot Product and Cross Product of Vectors
 Change of Basis of vectors is possible through transformations
 Eigen Vectors and Eigen Values involve matrix and matrix transformation.
 Abstract Vector Spaces also use the concepts of the transformation matrix.
Related Topics
The following topics help for a better understanding of the transformation matrix.
Examples on Transformation Matrix

Example 1: Find the new vector formed for the vector 5i + 4j, with the help of the transformation matrix \(\begin{bmatrix}2&3\\1&2\end{bmatrix}\).
Solution:
The given transformation matrix is T = \(\begin{bmatrix}2&3\\1&2\end{bmatrix}\)
The given vector A = 5i + 4j is written as a column matrix as A = \(\begin{bmatrix}5\\4\end{bmatrix}\)
Let the new matrix after transformation be B, and we have the transformation formul as TA = B
B = TA = \(\begin{bmatrix}2&3\\1&2\end{bmatrix}\) × \(\begin{bmatrix}5\\4\end{bmatrix}\)
B = \(\begin{bmatrix}2 ×5 + (3)×4\\1×5 + 2×4\end{bmatrix}\)
B = \(\begin{bmatrix}2\\13\end{bmatrix}\)
B = 2i + 13j
Therefore, the new matrix on transformation 2i + 13j.

Example 2: Find the value of the constant 'a' in the transformation matrix \(\begin{bmatrix}1&a\\0&1\end{bmatrix}\), which has transformed the vector A = 3i + 2j to another vector B = 7i + 2j
Solution:
The given vectors are A = 3i + 2j and B = 7i + 2j
These vectors written as column matrices is equal to A = \(\begin{bmatrix}3\\2\end{bmatrix}\), and B = \(\begin{bmatrix}7\\2\end{bmatrix}\)
This is a shear transformation, where only one component of the matrix is changes. The given transformation matrix is T = \(\begin{bmatrix}1&a\\0&1\end{bmatrix}\)
Applyig the formula of transformation matrix, TA = B, we have the following calculations.
\(\begin{bmatrix}1&a\\0&1\end{bmatrix}\) × \(\begin{bmatrix}3\\2\end{bmatrix}\) = \(\begin{bmatrix}7\\2\end{bmatrix}\)
\(\begin{bmatrix}1 ×3+ a × 2\\0 × 3 + 1 × 2\end{bmatrix}\) = \(\begin{bmatrix}7\\2\end{bmatrix}\)
\(\begin{bmatrix}3+ 2a \\ 2\end{bmatrix}\) = \(\begin{bmatrix}7\\2\end{bmatrix}\)
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 \(\begin{bmatrix}1&2\\0&1\end{bmatrix}\).
FAQs on Transformation Matrix
How Is Transformation Matrix?
Transformation Matrix is used to transform one vector into another vector by the process of matrix multiplication. The position vector of a point is represented as a column matrix, and the number of elements in this column matrix is equal to the components of the vector. The multiplication of a transformation matrix with the column matrix of the vector gives a new matrix of the transformed vector.
What Is the Transformation Matrix Formula?
The formula for transformation matrix is TA = A'. The transformation matrix T = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) on multiplication with a position vector A = xi + yj represented as a column matrix \(\begin{bmatrix}x\\y\end{bmatrix}\), transforms it into another matrix \(\begin{bmatrix}x\\y\end{bmatrix}\), representing a new matrix with position vector A' = x'i + y'j. The transformation matrix formula can be represented in the following matrix form.
\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) × \(\begin{bmatrix}x\\y\end{bmatrix}\) = \(\begin{bmatrix}x'\\y'\end{bmatrix}\)
How Does Transformation Matrix Work?
The transformation matrix works with the formula of matrix multiplication. The multiplication of the transformation matrix T = \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\), with a column matrix A = \(\begin{bmatrix}x\\y\end{bmatrix}\), results in a new matrri A' = \(\begin{bmatrix}x'\\y'\end{bmatrix}\). The condition for matrix multiplication should match before performing the multiplication of transformation matrix. The number of columns in the transformation matrix T should be equal to the number of elements or rows in the column matrix A.
What Are the Types of Transformation Matrix?
The type transformation matrix depends on the transformation which they can perform on the vector in a twodimensional or threedimensional space. The frequently performed transformations using a transformation matrix are stretching, squeezing, rotation, reflection, and orthogonal projection.
What Are the Uses of Transformation Matrix?
The following are some of the important uses of the transformation matrix.
 Vectors represented in a two or threedimensional frame are transformed to another vector.
 Linear Combinations of two or more vectors through multiplication are possible through a transformation matrix.
 The linear transformations of matrices can be used to change the matrices into another form.
 Matrix multiplication is the transformation of one matrix into another matrix.
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