1.9 More on Linear Transformations

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Section 1.9 More on Linear Transformations

Handout 1.9 More on Linear Transformation

Comic showing a humorous matrix transformation.

Objectives: Topics

The standard vectors and the standard matrix.

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Two and three dimensional transformations in more detail.

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Onto and one-to-one transformations.

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Objectives: Goals

Identify and construct linear transformations of a matrix.

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Characterize linear transformations as onto and/or one-to-one.

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Solve linear systems represented as linear transformations.

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Express linear transformations in other forms, such as matrix equations or vector equations.

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Definition 1.56. The Standard Vectors.

The standard vectors in $\mathbb{R}^n$ are the vectors $\vec e_1, \vec e_2, \ldots, \vec e_n\text{.}$

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They are:

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\begin{equation*} \vec e_1 = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \qquad \vec e_2 = \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, \qquad \ldots, \qquad \vec e_n = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}. \end{equation*}

For example, in $\mathbb{R}^3\text{:}$

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\begin{equation*} \vec e_1=\begin{bmatrix}1\\0\\0\end{bmatrix}, \qquad \vec e_2=\begin{bmatrix}0\\1\\0\end{bmatrix}, \qquad \vec e_3=\begin{bmatrix}0\\0\\1\end{bmatrix}. \end{equation*}

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Note: If $A$ is an $m\times n$ matrix with columns $\vec v_1,\vec v_2,\ldots,\vec v_n\text{,}$ then

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\begin{equation*} A\vec e_i = \vec v_i, \text{ for } i=1,2,\ldots,n. \end{equation*}

So multiplying a matrix by $\vec e_i$ gives column $i$ of $A\text{.}$

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Example 1.57. Example.

Compute:

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\begin{equation*} \begin{bmatrix} 1 \amp 2 \amp 3\\ 4 \amp 5 \amp 6\\ 7 \amp 8 \amp 9 \end{bmatrix}\vec e_2. \end{equation*}

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Theorem 1.58. The Standard Matrix of a Linear Transformation.

Let $T:\mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. Then there is a unique matrix $A$ such that

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\begin{equation*} T(\vec x)=A\vec x, \qquad \vec x\in \mathbb{R}^n. \end{equation*}

In fact, $A$ is an $m\times n$ matrix, and its $j$th column is $T(\vec e_j)\text{.}$ That is,

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\begin{equation*} A= \begin{bmatrix} T(\vec e_1) \amp T(\vec e_2) \amp \cdots \amp T(\vec e_n) \end{bmatrix}. \end{equation*}

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The matrix $A$ is the standard matrix for the linear transformation $T\text{.}$

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Example 1.59.

What is the linear transformation $T:\mathbb{R}^2 \to \mathbb{R}^2$ defined by “rotates vectors counterclockwise by angle $\theta$”?

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There is a long list of geometric transformations of $\mathbb{R}^2$ in many linear algebra texts (reflections, rotations, contractions and expansions, shears, projections, and more). Several are included at the end of this section.

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Definition 1.60. Onto.

A linear transformation $T:\mathbb{R}^n \to \mathbb{R}^m$ is onto if for every $\vec b\in \mathbb{R}^m$ there exists $\vec x\in \mathbb{R}^n$ such that $T(\vec x)=\vec b\text{.}$ In otherwords, a linear transformation is onto if the range of $T$ equals the codomain of $T\text{.}$

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Onto is an existence property: for any $\vec b\in\mathbb{R}^m\text{,}$ the system $A\vec x=\vec b$ has a solution (where $A$ is the standard matrix of $T$).

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Examples

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A rotation on the plane is onto.

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A projection in the plane is not onto.

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Theorem 1.61. Onto: Equivalent Statements.

For a linear transformation $T:\mathbb{R}^n \to \mathbb{R}^m$ with standard matrix $A\text{,}$ the following are equivalent:

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$T$ is onto.

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The columns of $A$ span $\mathbb{R}^m\text{.}$

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The matrix $A$ has a pivot in every row (equivalently, $m$ pivotal columns).

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Definition 1.62. One-to-One.

A linear transformation $T:\mathbb{R}^n \to \mathbb{R}^m$ is one-to-one if for every $\vec b\in\mathbb{R}^m$ there is at most one (possibly none) $\vec x\in\mathbb{R}^n$ such that $T(\vec x)=\vec b\text{.}$

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One-to-one is a uniqueness property; it does not assert existence for all $\vec b\text{.}$

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Examples

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A rotation on the plane is one-to-one.

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A projection in the plane is not one-to-one.

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Theorem 1.63. One-to-One: Equivalent Statements.

For a linear transformation $T:\mathbb{R}^n \to \mathbb{R}^m$ with standard matrix $A\text{,}$ the following are equivalent:

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$T$ is one-to-one.

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The unique solution to $T(\vec x)=\vec 0$ is $\vec x=\vec 0\text{.}$

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The columns of $A$ are linearly independent.

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Every column of $A$ is pivotal (equivalently, $A$ has no free variables).

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Example 1.64.

Complete the matrices below by entering numbers into the missing entries so that the stated properties are satisfied. If it isn’t possible, state why.

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(a)

$A$ is a $2\times 3$ standard matrix for a one-to-one linear transformation.

\begin{equation*} A= \begin{bmatrix} 1 \amp 0 \amp \\ 0 \amp \amp 1 \end{bmatrix}. \end{equation*}

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(b)

$B$ is a $3\times 2$ standard matrix for an onto linear transformation.

\begin{equation*} B= \begin{bmatrix} 1 \amp \\ \amp \\ \amp \end{bmatrix}. \end{equation*}

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(c)

$C$ is a $3\times 3$ standard matrix of a linear transformation that is one-to-one and onto.

\begin{equation*} C= \begin{bmatrix} 1 \amp 1 \amp 1\\ \amp \amp \\ \amp \amp \end{bmatrix}. \end{equation*}

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Example 1.65.

Define a linear transformation by

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\begin{equation*} T(x_1,x_2)=(3x_1+x_2,\; 5x_1+7x_2,\; x_1+3x_2). \end{equation*}

Is $T$ one-to-one? Is $T$ onto?

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Below are many examples of linear transformations from $\mathbb{R}^2$ to $\mathbb{R}^2\text{.}$

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transformation	image of unit square	standard matrix
reflection through $x_1$-axis	$Unit square reflected across the horizontal axis.$ Relection across $x_2 = 0$	$\begin{bmatrix}1\amp 0\\0\amp -1\end{bmatrix}$
reflection through $x_2$-axis	$Unit square reflected across the vertical axis.$ Relection across $x_1 = 0$	$\begin{bmatrix}-1\amp 0\\0\amp 1\end{bmatrix}$
reflection through $x_2=x_1$	$Unit square reflected across diagonal line passing through (0,0) and (1,1).$ Reflection across $x_2=x_1\text{.}$	$\begin{bmatrix}0\amp 1\\1\amp 0\end{bmatrix}$
reflection through $x_2=-x_1$	$Unit square reflected across the diagonal line passing through (0,0) and (1,-1).$ Reflection across $x_2=-x_1\text{.}$	$\begin{bmatrix}0\amp -1\\-1\amp 0\end{bmatrix}$

transformation	image of unit square	standard matrix
Horizontal contraction	$Unit square compressed horizontally toward the vertical axis.$ Horizontal contraction by factor $k$ with $\|k\|<1\text{.}$	$\begin{bmatrix}k\amp 0\\0\amp 1\end{bmatrix}\text{,}$ $\|k\|<1$
Horizontal expansion	$Unit square stretched horizontally away from the vertical axis.$ Horizontal expansion by factor $k$ with $k>1\text{.}$	$\begin{bmatrix}k\amp 0\\0\amp 1\end{bmatrix}\text{,}$ $k>1$
Vertical contraction	$Unit square compressed vertically toward the horizontal axis.$ Vertical contraction by factor $k$ with $\|k\|<1\text{.}$	$\begin{bmatrix}1\amp 0\\0\amp k\end{bmatrix}\text{,}$ $\|k\|<1$
Vertical expansion	$Unit square stretched vertically away from the horizontal axis.$ Vertical expansion by factor $k$ with $k>1\text{.}$	$\begin{bmatrix}1\amp 0\\0\amp k\end{bmatrix}\text{,}$ $k>1$

transformation	image of unit square	standard matrix
Horizontal shear (left)	$Unit square sheared left by a horizontal shift.$ Horizontal shear with parameter $k<0\text{.}$	$\begin{bmatrix}1\amp k\\0\amp 1\end{bmatrix}\text{,}$ $k<0$
Horizontal shear (right)	$Unit square sheared right by a horizontal shift.$ Horizontal shear with parameter $k>0\text{.}$	$\begin{bmatrix}1\amp k\\0\amp 1\end{bmatrix}\text{,}$ $k>0$
Vertical shear (down)	$Unit square sheared downward by a vertical shift.$ Vertical shear with parameter $k<0\text{.}$	$\begin{bmatrix}1\amp 0\\k\amp 1\end{bmatrix}\text{,}$ $k<0$
Vertical shear (up)	$Unit square sheared upward by a vertical shift.$ Vertical shear with parameter $k>0\text{.}$	$\begin{bmatrix}1\amp 0\\k\amp 1\end{bmatrix}\text{,}$ $k>0$

transformation	image of unit square	standard matrix
Projection onto the $x_1$-axis	$Unit square flattened onto the horizontal axis.$ Projection onto $x_1$-axis.	$\begin{bmatrix}1\amp 0\\0\amp 0\end{bmatrix}$
Projection onto the $x_2$-axis	$Unit square flattened onto the vertical axis.$ Projection onto $x_2$-axis.	$\begin{bmatrix}0\amp 0\\0\amp 1\end{bmatrix}$

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transformation	image of unit square	standard matrix
reflection through \(x_1\)-axis	$Unit square reflected across the horizontal axis.$ Relection across \(x_2 = 0\)	\(\begin{bmatrix}1\amp 0\\0\amp -1\end{bmatrix}\)
reflection through \(x_2\)-axis	$Unit square reflected across the vertical axis.$ Relection across \(x_1 = 0\)	\(\begin{bmatrix}-1\amp 0\\0\amp 1\end{bmatrix}\)
reflection through \(x_2=x_1\)	$Unit square reflected across diagonal line passing through (0,0) and (1,1).$ Reflection across \(x_2=x_1\text{.}\)	\(\begin{bmatrix}0\amp 1\\1\amp 0\end{bmatrix}\)
reflection through \(x_2=-x_1\)	$Unit square reflected across the diagonal line passing through (0,0) and (1,-1).$ Reflection across \(x_2=-x_1\text{.}\)	\(\begin{bmatrix}0\amp -1\\-1\amp 0\end{bmatrix}\)

transformation	image of unit square	standard matrix
Horizontal contraction	$Unit square compressed horizontally toward the vertical axis.$ Horizontal contraction by factor \(k\) with \(\|k\|<1\text{.}\)	\(\begin{bmatrix}k\amp 0\\0\amp 1\end{bmatrix}\text{,}\) \(\|k\|<1\)
Horizontal expansion	$Unit square stretched horizontally away from the vertical axis.$ Horizontal expansion by factor \(k\) with \(k>1\text{.}\)	\(\begin{bmatrix}k\amp 0\\0\amp 1\end{bmatrix}\text{,}\) \(k>1\)
Vertical contraction	$Unit square compressed vertically toward the horizontal axis.$ Vertical contraction by factor \(k\) with \(\|k\|<1\text{.}\)	\(\begin{bmatrix}1\amp 0\\0\amp k\end{bmatrix}\text{,}\) \(\|k\|<1\)
Vertical expansion	$Unit square stretched vertically away from the horizontal axis.$ Vertical expansion by factor \(k\) with \(k>1\text{.}\)	\(\begin{bmatrix}1\amp 0\\0\amp k\end{bmatrix}\text{,}\) \(k>1\)

transformation	image of unit square	standard matrix
Horizontal shear (left)	$Unit square sheared left by a horizontal shift.$ Horizontal shear with parameter \(k<0\text{.}\)	\(\begin{bmatrix}1\amp k\\0\amp 1\end{bmatrix}\text{,}\) \(k<0\)
Horizontal shear (right)	$Unit square sheared right by a horizontal shift.$ Horizontal shear with parameter \(k>0\text{.}\)	\(\begin{bmatrix}1\amp k\\0\amp 1\end{bmatrix}\text{,}\) \(k>0\)
Vertical shear (down)	$Unit square sheared downward by a vertical shift.$ Vertical shear with parameter \(k<0\text{.}\)	\(\begin{bmatrix}1\amp 0\\k\amp 1\end{bmatrix}\text{,}\) \(k<0\)
Vertical shear (up)	$Unit square sheared upward by a vertical shift.$ Vertical shear with parameter \(k>0\text{.}\)	\(\begin{bmatrix}1\amp 0\\k\amp 1\end{bmatrix}\text{,}\) \(k>0\)

transformation	image of unit square	standard matrix
Projection onto the \(x_1\)-axis	$Unit square flattened onto the horizontal axis.$ Projection onto \(x_1\)-axis.	\(\begin{bmatrix}1\amp 0\\0\amp 0\end{bmatrix}\)
Projection onto the \(x_2\)-axis	$Unit square flattened onto the vertical axis.$ Projection onto \(x_2\)-axis.	\(\begin{bmatrix}0\amp 0\\0\amp 1\end{bmatrix}\)