A linear transformation is a function that maps an input vector to an output vector in such a way that it maintains the linear structure of the input space. This means that the transformation preserves vector addition and scalar multiplication operations.

Mathematically, a linear transformation L from a vector space \Reals ^{2} to another vector space \Reals ^{2} can be expressed as: L(\overrightarrow{u} + \overrightarrow{v}) = L(\overrightarrow{u}) + L(\overrightarrow{v}) and L(c\overrightarrow{v}) = cL(\overrightarrow{v}) where u and v are vectors in \Reals^{2}, and c is a scalar. It is important to note that each component of the output vector is the sum of constant multiples of the input vector components.

An important property of linear transformations is that they map the zero vector to the zero vector.

In other words, L(0) = 0, where 0 is the zero vector.

Vectors in \Reals^{2} can be represented using matrices. A single vector can be expressed as a 2 \times 1 matrix, while a set of n vectors can be expressed as a 2 \times n matrix.

For example, if we have a vector v = [x,\, y], it can be represented as a 2 \times 1 matrix: v=\begin{bmatrix} x \\ y \end{bmatrix}.

And if we have a set of three vectors v_{1},\,v_{2},\, and v_{3}, they can be represented as a 2 \times 3 matrix: v=\begin{bmatrix} x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{bmatrix}

Examples

Example 1

Given a linear transformation L that is represented by the matrix v=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, find the output vector w when the input vector v is [1,\, 1].

Worked Solution

Create a strategy

We know that the output vector w is obtained by multiplying the transformation matrix A with the input vector v:

w = A \times v

Apply the idea

\displaystyle w	\displaystyle =	\displaystyle \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times [1,\, 1]	Substitute the given matrix and vector into the equation
\displaystyle w	\displaystyle =	\displaystyle \begin{bmatrix} 1(1) + 2(1) & 3(1) + 4(1) \end{bmatrix}	Multiply the matrix by the vector
\displaystyle w	\displaystyle =	\displaystyle [3,\,7]	Evaluate

So, the output vector w = [3,\, 7].

Example 2

Let's consider a linear transformation L represented by the matrix v=\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}. Find the output vector when the input vector is [4,\, 4].

Worked Solution

Create a strategy

Use the linear transformation rule: w = A \times v.

Apply the idea

\displaystyle w	\displaystyle =	\displaystyle \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \times [4,\, 4]	Substitute the given matrix and vector into the equation
\displaystyle w	\displaystyle =	\displaystyle \begin{bmatrix} 2(4) + 0(4) & 0(4) + 2(4) \end{bmatrix}	Multiply the matrix by the vector
\displaystyle w	\displaystyle =	\displaystyle [8,\,8]	Evaluate

So, the output vector w = [8,\, 8].

Idea summary

A linear transformation is a function that maps input vectors to output vectors, and in \Reals^{2}, a unique 2 \times 2 matrix A represents this transformation. Linear transformations map the zero vector to the zero vector, and a single vector in \Reals^{2} can be expressed as a \\2 \times 1 matrix.

Linear transformations and matrices

For a linear transformation, L, from \Reals ^{2} to \Reals ^{2}, there is a unique 2 \times 2 matrix, A, such that \\L(\overrightarrow{v}) = A (\overrightarrow{v}) for all vectors in \Reals^{2}. Conversely, for a given 2 \times 2 matrix, A, the function \\L(\overrightarrow{v}) = A (\overrightarrow{v}) is a linear transformation from R^{2} to R^{2}. This matrix A is known as the transformation matrix.

To find the output vectors for a linear transformation L(\overrightarrow{v}) = A (\overrightarrow{v}), we multiply the 2 \times 2 transformation matrix, A, with the 2 \times n matrix of n input vectors. This results in a 2 \times n matrix of the n output vectors for the linear transformation.

For example, let A be the transformation matrix: A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

And let V be the matrix of input vectors: v = \begin{bmatrix} x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{bmatrix}

Then, the matrix of output vectors, W, is given by: W =\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{bmatrix}

Examples

Example 3

Let's say we have a linear transformation represented by the matrix A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}. We want to find the output vector when the input vector is v = \begin{bmatrix} 5 \\ 1 \end{bmatrix}.

Worked Solution

Create a strategy

To find the output vector, we perform matrix multiplication, Av.

Apply the idea

\displaystyle Av	\displaystyle =	\displaystyle \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \times \begin{bmatrix} 5 \\ 1 \end{bmatrix}	Substitute the given matrix and vector into the equation
	\displaystyle =	\displaystyle \begin{bmatrix} 2(5) + 3(1) \\ 1(5) + 4(1) \end{bmatrix}	Multiply the matrix by the vector
	\displaystyle =	\displaystyle \begin{bmatrix} 13 \\ 9 \end{bmatrix}	Evaluate

So, the output vector is \begin{bmatrix} 13 \\ 9 \end{bmatrix}.

Example 4

Consider the linear transformation represented by the matrix B=\begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix}. We want to find the output vector when the input vector is u =\begin{bmatrix} 2 \\ 3 \end{bmatrix}.

Worked Solution

Create a strategy

To find the output vector, we perform matrix multiplication, Bu.

Apply the idea

\displaystyle Bu	\displaystyle =	\displaystyle \begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix} \times \begin{bmatrix} 2 \\ 3 \end{bmatrix}	Substitute the given matrix and vector into the equation
	\displaystyle =	\displaystyle \begin{bmatrix} -1(2) + 2(3) \\ 0(2) + -1(3) \end{bmatrix}	Multiply the matrix by the vector
	\displaystyle =	\displaystyle \begin{bmatrix} 4 \\ -3 \end{bmatrix}	Evaluate

So, the output vector is \begin{bmatrix} 13 \\ 9 \end{bmatrix}.

Idea summary

For a linear transformation L(v) = Av, the output vectors can be found by multiplying the transformation matrix A with the matrix of input vectors. This results in a matrix of output vectors, where each component of the output vectors is the sum of constant multiples.

4.12 Linear transformations and matrices

Introduction

Ideas

Understanding linear transformations

Examples

Example 1

Example 2

Linear transformations and matrices

Examples

Example 3

Example 4

Outcomes

4.12.A

What is Mathspace

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