Objectives: Topics
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The definition of a linear transformation.
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Interpreting matrix multiplication as a linear transformation.
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Section 1.8 An Introduction to Linear Transformations
Handout 1.8 An Introduction to Linear Transformations
Objectives: Goals
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Construct and interpret linear transformations in \(\mathbb{R}^n\) (e.g., view a linear transformation as a projection or a shear).
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Characterize linear transformations using:
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existence and uniqueness,
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domain, codomain, and range.
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Definition 1.49.
\begin{equation*}
T:\mathbb{R}^n \to \mathbb{R}^m, \qquad T(\vec v) = A\vec v .
\end{equation*}
A function of this form is called a matrix transformation.
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The domain of \(T\) is \(\mathbb{R}^n\text{.}\)
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The codomain (or target) of \(T\) is \(\mathbb{R}^m\text{.}\)
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The vector \(T(\vec x)\) is the image of \(\vec x\) under \(T\text{.}\)
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The set of all possible images \(T(\vec x)\) is the range of \(T\text{.}\)
This yields another interpretation of the equation \(A\vec x=\vec b\text{:}\)
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as a system of linear equations,
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as an augmented matrix,
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as a vector equation,
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as a matrix equation,
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as a linear transformation equation.
Many familiar functions have domain and codomain \(\mathbb{R}\text{.}\) For example, the sine function can be described by the rule
\begin{equation*}
f\colon \mathbb{R}\to\mathbb{R}, \qquad f(x)=\sin(x).
\end{equation*}
In calculus we often visualize a function via its graph: the horizontal axis represents the domain and the vertical axis the codomain.
Graph of \(y = \sin(x)\) on the interval \([-2\pi, 2\pi]\) with axes labeled \(x\) and \(y\) and a light grid.
Graphing is fine when both domain and codomain are \(\mathbb{R}\text{.}\) If the domain is \(\mathbb{R}^2\) and the codomain is \(\mathbb{R}^3\text{,}\) we would need five dimensions to draw the full graph, so a geometric graph is not possible.
Example 1.50.
Let \(A=\begin{bmatrix} 1\amp 1\\[2pt] 0\amp 1\\[2pt] 1\amp 1 \end{bmatrix},\qquad
\vec u=\begin{bmatrix} 3\\[2pt] 4 \end{bmatrix},\qquad
\vec b=\begin{bmatrix} 7\\[2pt] 5\\[2pt] 7 \end{bmatrix}.\)
Consider the matrix transformation \(T(\vec v)=A\vec v\text{.}\)
(a)
Compute \(T(\vec u)\text{.}\)
(b)
(c)
Give a vector \(\vec c\in\mathbb{R}^3\) for which there is no \(\vec v\) with \(T(\vec v)=\vec c\text{.}\)
Equivalently: Give a \(\vec c\) not in the range of \(T\text{,}\) i.e., not in the span of the columns of \(A\text{.}\)
Definition 1.51.
A function \(T:\mathbb{R}^n\to\mathbb{R}^m\) is linear if for all \(\vec u,\vec v\in\mathbb{R}^n\) and all scalars \(c\in\mathbb{R}\text{:}\)
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\(T(\vec u+\vec v)=T(\vec u)+T(\vec v)\text{,}\)
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\(T(c\,\vec v)=c\,T(\vec v)\text{.}\)
By linearity, for any scalars \(c_1,\dots,c_k\) and vectors \(\vec v_1,\dots,\vec v_k\text{,}\)
\begin{equation*}
T\!\big(c_1\vec v_1+\cdots+c_k\vec v_k\big)
\;=\;
c_1\,T(\vec v_1)+\cdots+c_k\,T(\vec v_k).
\end{equation*}
This is the principle of superposition. In particular, once we know \(T(\vec e_1),\dots,T(\vec e_n)\text{,}\) we can compute \(T(\vec v)\) for any \(\vec v \in \mathbb{R}^n\text{.}\)
Fact 1.52.
Every matrix transformation \(T_A(\vec v)=A\vec v\) is linear.
Example 1.53.
Suppose \(T(\vec x)=A\vec x\text{.}\) Give a brief geometric description of the action of \(T\) in \(\mathbb{R}^2\) for each matrix.
(a)
\(A=\begin{bmatrix} 0\amp 1\\[2pt] 1\amp 0 \end{bmatrix}\)
(b)
\(A=\begin{bmatrix} 1\amp 0\\[2pt] 0\amp 0 \end{bmatrix}\)
(c)
Example 1.54.
(a)
\(A=\begin{bmatrix} 1\amp 0\amp 0\\[2pt] 0\amp 1\amp 0\\[2pt] 0\amp 0\amp 0 \end{bmatrix}\)
(b)
\(A=\begin{bmatrix} 1\amp 0\amp 0\\[2pt] 0\amp -1\amp 0\\[2pt] 0\amp 0\amp 1 \end{bmatrix}\)
Example 1.55.
A linear transformation \(T:\mathbb{R}^2\to\mathbb{R}^3\) satisfies
\begin{equation*}
T\!\left(\begin{bmatrix}1\\[2pt]0\end{bmatrix}\right)
=
\begin{bmatrix}5\\[2pt]-7\\[2pt]2\end{bmatrix},
\qquad
T\!\left(\begin{bmatrix}0\\[2pt]1\end{bmatrix}\right)
=
\begin{bmatrix}-3\\[2pt]8\\[2pt]0\end{bmatrix}.
\end{equation*}
What is the matrix that represents \(T\text{?}\)
