What is a Linear Transformation?

It is a function (the word ‘transformation‘ means the same thing as the word ‘function‘) which takes vectors as inputs and produces vectors as outputs. Linear transformation is also called ‘Linear Map‘ or ‘Linear Mapping‘.
A transformation T that takes n-dimensional vectors as inputs and produces m-dimensional vectors as outputs is
denoted as, T: \mathbb{R}^{n} \to \mathbb{R}^{m} . This is the mapping notation of a linear transformation.

How is a linear transformation defined mathematically?

A linear transformation is a function T: \mathbb{R}^{n} \to \mathbb{R}^{m} that satisfies the following properties for any vectors \left \{ \vec{u}, \vec{v} \in \mathbb{R}^{n}\right \} and any scalars \left \{c, d \in \mathbb{R}\right \}
T\left ( c\vec{v} \right )= cT\left ( \vec{v} \right )
T\left ( \vec{u} + \vec{v} \right ) = T\left ( \vec{u} \right ) + T\left ( \vec{v} \right )
The above two properties can be combined into a single statement:
T\left ( c\vec{u} + d\vec{v} \right ) = cT\left ( \vec{u} \right ) + dT\left ( \vec{v} \right )

How do I understand this mathematical definition of linear transformation?

The matrix-vector product is an implementation of the abstract definition of linear transformation.
Multiplying a matrix Mm x n with a vector vn x 1 outputs a vector wm x 1.
This can be represented as T\left ( \vec{v} \right )= M\vec{v}
For example, Let M be a 2×3 matrix defined as,
M = \begin{bmatrix} 1 & 0 &-1 \\ 3& 1 &2 \end{bmatrix}
and v be a 3×1 vector defined as,
\vec{v} = \begin{bmatrix} x\\ y\\ z \end{bmatrix}
Multiplying M with v results in a 2×1 vector
\begin{bmatrix} 1 & 0 &-1 \\ 3& 1 &2 \end{bmatrix} \begin{bmatrix} x\\ y\\ z \end{bmatrix} = \begin{bmatrix} x-z\\ 3x+y+2z\end{bmatrix}
Thus, T\left ( \vec{v} \right )= \begin{bmatrix} x-z\\ 3x+y+2z\end{bmatrix} is a function which takes a 3-dimensional vector as an input and outputs a 2-dimensional vector. This can be denoted by the mapping notation as, T: \mathbb{R}^{3} \to \mathbb{R}^{2}
As an example, T\begin{bmatrix} 2\\ 4\\ 6 \end{bmatrix}=\begin{bmatrix} -4\\ 22 \end{bmatrix}

But how does the function T\left ( \vec{v} \right )= \begin{bmatrix} x-z\\ 3x+y+2z\end{bmatrix} conform to the mathematical definition of linear transformation?

In order to be a classified as a linear transformation, the function T\left ( \vec{v} \right )= \begin{bmatrix} x-z\\ 3x+y+2z\end{bmatrix} where \vec{v} is 3-dimensional input vector, should satisfy the above two properties of linearity for any 3-dimensional vector.
Lets say \vec{a} and \vec{b} are 3-dimensional vectors such that,
\vec{a}=\begin{bmatrix} x_{1}\\y_{1} \\z_{1} \end{bmatrix}, \vec{b}=\begin{bmatrix} x_{2}\\y_{2} \\z_{2} \end{bmatrix}
Proof for property 1:
Pass the \vec{a} as an input to the transformation function T\left ( \vec{v} \right ) ,
T\left ( c\vec{a} \right ) = c\begin{bmatrix} x_{1} - z_{1}\\ 3x_{1}+y_{1}+2z_{1} \end{bmatrix}
T\left ( c\vec{a} \right ) = \begin{bmatrix} cx_{1} - cz_{1}\\ 3cx_{1}+cy_{1}+2cz_{1} \end{bmatrix}
Also,

cT\left ( \vec{a} \right ) = c\begin{bmatrix} x_{1} - z_{1}\\ 3x_{1}+y_{1}+2z_{1} \end{bmatrix}
cT\left ( \vec{a} \right ) = \begin{bmatrix} cx_{1} - cz_{1}\\ 3cx_{1}+cy_{1}+2cz_{1} \end{bmatrix}
Hence, we can see that, T\left ( c\vec{a} \right )=cT\left ( \vec{a} \right )

Proof for property 2:
Pass the sum \vec{a} + \vec{b} to the transformation function T\left ( \vec{v} \right ) ,
T\left ( \vec{a} + \vec{b} \right ) = \begin{bmatrix} \left ( x_{1}+x_{2} \right ) -\left ( z_{1}+z_{2} \right )\\ 3\left ( x_{1}+x_{2} \right )+\left ( y_{1}+y_{2} \right )+2\left ( z_{1}+z_{2} \right ) \end{bmatrix}
T\left ( \vec{a} + \vec{b} \right ) = \begin{bmatrix} x_{1}+x_{2}-z_{1}-z_{2}\\ 3x_{1}+3x_{2}+y_{1}+y_{2}+2z_{1}+2z_{2} \end{bmatrix}
Also,
T\left ( \vec{a} \right ) + T\left ( \vec{b} \right ) = \begin{bmatrix} x_{1}-z_{1}\\ 3x_{1}+y_{1}+2z_{1} \end{bmatrix} + \begin{bmatrix} x_{2}-z_{2}\\ 3x_{2}+y_{2}+2z_{2} \end{bmatrix}
T\left ( \vec{a} \right ) + T\left ( \vec{b} \right ) = \begin{bmatrix} x_{1}+x_{2}-z_{1}-z_{2}\\ 3x_{1}+3x_{2}+y_{1}+y_{2}+2z_{1}+2z_{2} \end{bmatrix}

Therefore, we can see that, T\left ( \vec{a}+\vec{b} \right )=T\left ( \vec{a} \right )+T\left ( \vec{b} \right )

Is there an easy way to identify that a function is a linear transformation?

Examine term of each component of f\left ( x \right ) . If each of these terms is a number times one of the components of x, then f is a linear transformation. For example, the function f\left ( x,y,z \right )=\left ( 3x-y,3z,0,z-2x \right ) is a linear transformation.
But neither g\left ( x,y,z \right )=\left ( 3x-y,3z+2,0,z-2x \right ) nor h\left ( x,y,z \right )=\left ( 3x-y,3xz,0,z-2x \right ) are linear transformations. The reason is that the function g has a component 3z+2 with the term 2 which is a constant and does not contain any components of our input vector (x,y,z). And the function h has a nonlinear component 3xz which is a product of two components x and z.

How can I undo the effects of a linear transformation?

Multiplying the transformed vector with the inverse of the transformation matrix will yield the input vector.
But this ‘undoing‘ the effects of a transformation only works if the transformation matrix is invertible.

As an example, if the transformation matrix is,
M=\begin{bmatrix} 1 &2 \\ 0 &3 \end{bmatrix}
and input vector is,
\vec{v}=\begin{bmatrix} x_{1}\\ y_{1} \end{bmatrix}
Linear transformation is Mv. Therefore,
M\vec{v}=\begin{bmatrix} 1 &2 \\ 0 &3 \end{bmatrix} \begin{bmatrix} x_{1}\\y_{1} \end{bmatrix} =\begin{bmatrix} x_{1}+2y_{1}\\ 3y_{1} \end{bmatrix}
To get back the input vector v, multiply by M-1

M^{-1}\left ( M\vec{v} \right )= \begin{bmatrix} 1 &-2/3 \\ 0 &1/3 \end{bmatrix} \begin{bmatrix} 1 &2 \\ 0 &3 \end{bmatrix} \begin{bmatrix} x_{1}\\y_{1} \end{bmatrix} = \begin{bmatrix} 1 &-2/3 \\ 0 &1/3 \end{bmatrix} \begin{bmatrix} x_{1}+2y_{1}\\ 3y_{1} \end{bmatrix} = \begin{bmatrix} x_{1}+2y_{1}-2y_{1}\\ y_{1} \end{bmatrix}
M^{-1}\left ( M\vec{v} \right )= \begin{bmatrix} x_{1}\\ y_{1} \end{bmatrix}

Why is a Linear transformation called “Linear”?

Linear transformation output has two important properties:

  1. All lines remain lines and do not turn into a curve after the transformation (probably that’s the
    reason it’s called ‘linear’) and,
  2. The origin always stays fixed and does not change after the transformation.

I hope this post clarifies the concepts of Linear Transformations. See you in the next blog!

Introduction to Linear Transformations

Related posts:

  1. How to find the transformation matrix of a linear transformation
  2. How to create a vector in Python using numpy
  3. What is the significance of matrix multiplication
  4. How to translate between coordinate systems that use different basis vectors
    

One thought on “Introduction to Linear Transformations”

  • Anil kumar
    June 20, 2018 at 10:04 AM
    Permalink

    Superb

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