Linear Transformation is a interesting concept of linear algebra which is a function of two vector spaces which fulfills the following properties:

Additivity: T(a + s) = T(a) + T(s)

Homogeneity: T(k • a) = k • T(a)

Lets say there's a Transformation function T which takes a 2D vector and returns a 2D vector.

T([p, q]) = [2p + q, p - q]

Lets take a vector [1, 2] and apply the transformation function T on it.

T([1, 2]) = [2(1) + 2, 1 - 2]

T([1, 2]) = [4, -1]

Now lets take another vector [3, 4] and apply the transformation function T on it.

T([3, 4]) = [2(3) + 4, 3 - 4]

T([3, 4]) = [10, -1]

Now lets take a vector [1, 2] and another vector [3, 4] and add them.

T([1, 2] + [3, 4]) = T([4, 6])

T([4, 6]) = [2(4) + 6, 4 - 6]

T([4, 6]) = [14, -2]

Now lets apply the transformation function T on [1, 2] and [3, 4] separately and add them.

T([1, 2]) + T([3, 4]) = [4, -1] + [10, -1]

T([1, 2]) + T([3, 4]) = [14, -2]

The results are same. Hence the function T is a linear transformation.

This function is a linear transformation because it satisfies the properties of linearity.

Linear Transformation is widely used in computer graphics. Mostly for rendering 3D objects on 2D screens.

It is also used in Machine learning for feature scaling and normalization.

The matrix linear transformation tool above is a simple yet powerful transformation calculator, which allows you to test your linear transformation functions.

You can input the transformation matrix and the vector on which you want to apply the transformation.

The calculator will then apply the transformation on the vector and show the result.

There is a setting button on the top right corner of the calculator which allows you to change the dimensions of the transformation matrix