Linear Transformations

Section 17.4 Linear Transformations

There are many applications of matrices where we view the matrix as a transformation (or mapping) that takes one vector and transforms (or maps) it to another vector. So, if

A

is an

m \times n

matrix then

A

can be thought of as a transformation that takes the

n \times 1

vector

x

to the

m \times 1

vector

b = A x .

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Example 17.4.1.

Consider the matrix

A = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) .

Since $A$ is a $2 \times 2$ matrix, we can think of $A$ as a transformation that takes each vector $x$ in the plane to another vector in the plane, $b = A x .$ Under this transformation, for example, $x = (\begin{matrix} 2 \\ 1 \end{matrix})$ goes to

b = A x = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} 2 \\ 1 \end{matrix}) = (\begin{matrix} 1 \\ 2 \end{matrix})

and $x = (\begin{matrix} - 1 \\ - 4 \end{matrix})$ goes to

b = A x = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} - 1 \\ 4 \end{matrix}) = (\begin{matrix} - 4 \\ - 1 \end{matrix}) .

See Figure 17.4.2

In fact, $A$ is the matrix for the transformation that we would describe as a reflection in the line $y = x .$ To see this, let the vector $(x, y)$ be transformed to the vector $(b_{1}, b_{2})$ under this reflection. Then we know that

\begin{aligned} b_{1} & = y = 0 x + y \\ b_{2} & = x = 1 x + 0 y \end{aligned}

which we can write via matrices as

b = (\begin{matrix} b_{1} \\ b_{2} \end{matrix}) = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = A x .

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Example 17.4.3.

It can be shown that a rotation (in the plane) about the origin through angle $θ$ can be represented by the matrix

A = (\begin{matrix} \cos (θ) & - \sin (θ) \\ \sin (θ) & \cos (θ) \end{matrix}) .

For example the matrix for a rotation about the origin through $π^{c}$ (or $180^{\circ}$ ) is

A = (\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) .

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Example 17.4.4.

Determine the matrix for the transformation in which a reflection in the line $y = x$ is followed by a rotation about the origin through $π^{c} .$

Answer.

(\begin{matrix} 0 & - 1 \\ - 1 & 0 \end{matrix})

Solution.

Let $A_{1} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$ and $A_{2} = (\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) .$ Then, under the given transformation the vector $x$ will be mapped to the vector $b$ where

b = A_{2} (A_{1} x) = (A_{2} A_{1}) x,

so the required matrix $A$ can be found via matrix multiplication, i.e.,

A = A_{2} A_{1} = (\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) = (\begin{matrix} 0 & - 1 \\ - 1 & 0 \end{matrix}) .

Note that given that the matrix for a reflection in the line $y = k x$ is

(\begin{matrix} \frac{1 - k^{2}}{1 + k^{2}} & \frac{2 k}{1 + k^{2}} \\ \frac{2 k}{1 + k^{2}} & - \frac{1 - k^{2}}{1 + k^{2}} \end{matrix})

we can see that the matrix $A$ represents a reflection in the line $y = - x .$

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Transformations that can be represented via a matrix are called linear transformations. When the matrix is a

2 \times 2

matrix the transformation is called a linear transformation of the plane.

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Exercises Example Tasks

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1.

Which vectors are mapped to ${(\begin{matrix} 1 & 1 & 0 \end{matrix})}^{T}$ under the transformation whose matrix is

(\begin{matrix} 1 & 2 & - 1 \\ 0 & - 1 & 2 \\ 2 & 3 & 0 \end{matrix}) t e x t .

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2.

Find the matrix for the transformation of the plane in which a rotation about the origin through $180^{\circ}$ is followed by a reflection in the line $y = 2 x .$

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3.

Find the vector to which $x = (\begin{matrix} 1 \\ 2 \end{matrix})$ is mapped by $(\begin{matrix} 3 & 4 \\ 1 & 2 \end{matrix})$ followed by $(\begin{matrix} 2 & 1 \\ 3 & 3 \end{matrix})$ followed by $(\begin{matrix} - 1 & 1 \\ - 1 & 0 \end{matrix}) .$