Learning Outcomes
- Translate back and forth between a linear transformation of Euclidean spaces and its standard matrix, and perform related computations.
Section 20.2 Standard Matrices (AT2)
Subsection 20.2.1 Warm Up
Remark 20.16.
Recall that a linear map \(T:V\rightarrow W\) satisfies
- \(T(\vec{v}+\vec{w}) = T(\vec{v})+T(\vec{w})\) for any \(\vec{v},\vec{w} \in V\text{.}\)
- \(T(c\vec{v}) = cT(\vec{v})\) for any \(c \in \IR,\vec{v} \in V\text{.}\)
In other words, a map is linear when vector space operations can be applied before or after the transformation without affecting the result.
Activity 20.17.
Can you recall the following?
(a)
Given a transformation, what do the terms domain and codomain mean?
(b)
What does the notation \(T\colon V\to W\) mean?
Subsection 20.2.2 Class Activities
Activity 20.18.
Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
\text{.}\) What is \(T\left(\left[\begin{array}{c} 3 \\ 0 \\ 0 \end{array}\right]\right)\text{?}\)
- \(\displaystyle \left[\begin{array}{c} 6 \\ 3\end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} -9 \\ 6 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} -4 \\ -2 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} 6 \\ -4 \end{array}\right]\)
Activity 20.19.
Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
\text{.}\) What is \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right]\right)\text{?}\)
- \(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)
Activity 20.20.
Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
\text{.}\) What is \(T\left(\left[\begin{array}{c} -2 \\ 0 \\ -3 \end{array}\right]\right)\text{?}\)
- \(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)
Activity 20.21.
Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
\text{.}\) What piece of information would help you compute \(T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)\text{?}\)
- The value of \(T\left(\left[\begin{array}{c} 0\\4\\0\end{array}\right]\right)= \left[\begin{array}{c} -4 \\ 16\end{array}\right]\text{.}\)
- The value of \(T\left(\left[\begin{array}{c} 0\\1\\0\end{array}\right]\right)= \left[\begin{array}{c} -1 \\ 4\end{array}\right]\text{.}\)
- The value of \(T\left(\left[\begin{array}{c} 1\\1\\1\end{array}\right]\right)= \left[\begin{array}{c} -2 \\ 7\end{array}\right]\text{.}\)
- Any of the above.
Observation 20.22.
Since all three choices in Activity 20.21 create a spanning and linearly independent set along with \(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right]\) and \(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right]\text{,}\) they each may be used to compute \(T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)\text{:}\)
\begin{equation*}
T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)
=
T\left(\left[\begin{array}{c}0\\4\\0\end{array}\right]\right)
-
T\left(\left[\begin{array}{c}0\\0\\1\end{array}\right]\right)
=
\left[\begin{array}{c} -4 \\ 16\end{array}\right]
-
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
=
\left[\begin{array}{c} -1 \\ 14 \end{array}\right]
\end{equation*}
\begin{equation*}
T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)
=
4
T\left(\left[\begin{array}{c}0\\1\\0\end{array}\right]\right)
-
T\left(\left[\begin{array}{c}0\\0\\1\end{array}\right]\right)
=
4
\left[\begin{array}{c} -1 \\ 4\end{array}\right]
-
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
=
\left[\begin{array}{c} -1 \\ 14 \end{array}\right]
\end{equation*}
\begin{equation*}
T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)
=
4
T\left(\left[\begin{array}{c}1\\1\\1\end{array}\right]\right)
-5
T\left(\left[\begin{array}{c}0\\0\\1\end{array}\right]\right)
-4
T\left(\left[\begin{array}{c}1\\0\\0\end{array}\right]\right)
\end{equation*}
\begin{equation*}
=
4
\left[\begin{array}{c} -2 \\ 7\end{array}\right]
-5
\left[\begin{array}{c} -3 \\ 2 \end{array}\right]
-4
\left[\begin{array}{c} 2 \\ 1 \end{array}\right]
=
\left[\begin{array}{c} -8+15-8 \\ 28-10-4 \end{array}\right]
=
\left[\begin{array}{c} -1 \\ 14 \end{array}\right]
\end{equation*}
Fact 20.23.
Consider any basis \(\{\vec b_1,\dots,\vec b_n\}\) for \(V\text{.}\) Since every vector \(\vec v\) can be written as a linear combination of basis vectors, \(\vec v = x_1\vec b_1+\dots+ x_n\vec b_n\text{,}\) we may compute \(T(\vec v)\) as follows:
\begin{equation*}
T(\vec v)=T(x_1\vec b_1+\dots+ x_n\vec b_n)=
x_1T(\vec b_1)+\dots+x_nT(\vec b_n)
.
\end{equation*}
Therefore any linear transformation \(T:V \rightarrow W\) can be defined by just describing the values of \(T(\vec b_i)\text{.}\)
Put another way, the images of the basis vectors completely determine the transformation \(T\text{.}\)
Definition 20.24.
Since a linear transformation \(T:\IR^n\to\IR^m\) is determined by its action on the standard basis \(\{\vec e_1,\dots,\vec e_n\}\text{,}\) it is convenient to store this information in an \(m\times n\) matrix, called the standard matrix of \(T\text{,}\) given by \([T(\vec e_1) \,\cdots\, T(\vec e_n)]\text{.}\)
For example, let \(T: \IR^3 \rightarrow \IR^2\) be the linear map determined by the following values for \(T\) applied to the standard basis of \(\IR^3\text{.}\)
\begin{equation*}
\scriptsize
T\left(\vec e_1 \right)
=
T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} 2 \\ 1\end{array}\right]
\hspace{2em}
T\left(\vec e_2 \right)
=
T\left(\left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right] \right)
=
\left[\begin{array}{c} -1 \\ 4\end{array}\right]
\hspace{2em}
T\left(\vec e_3 \right)
=
T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right)
=
\left[\begin{array}{c} -3 \\ 2\end{array}\right]
\end{equation*}
Then the standard matrix corresponding to \(T\) is
\begin{equation*}
\left[\begin{array}{ccc}T(\vec e_1) & T(\vec e_2) & T(\vec e_3)\end{array}\right]
=
\left[\begin{array}{ccc}2 & -1 & -3 \\ 1 & 4 & 2 \end{array}\right]
.
\end{equation*}
Activity 20.25.
Let \(T: \IR^4 \rightarrow \IR^3\) be the linear transformation given by
\begin{equation*}
T\left(\vec e_1 \right)
=
\left[\begin{array}{c} 0 \\ 3 \\ -2\end{array}\right]
\hspace{2em}
T\left(\vec e_2 \right)
=
\left[\begin{array}{c} -3 \\ 0 \\ 1\end{array}\right]
\hspace{2em}
T\left(\vec e_3 \right)
=
\left[\begin{array}{c} 4 \\ -2 \\ 1\end{array}\right]
\hspace{2em}
T\left(\vec e_4 \right)
=
\left[\begin{array}{c} 2 \\ 0 \\ 0\end{array}\right]
\end{equation*}
Write the standard matrix \([T(\vec e_1) \,\cdots\, T(\vec e_n)]\) for \(T\text{.}\)
Activity 20.26.
Let \(T: \IR^3 \rightarrow \IR^2\) be the linear transformation given by
\begin{equation*}
T\left(\left[\begin{array}{c} x\\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} x+3z \\ 2x-y-4z \end{array}\right]
\end{equation*}
(a)
Compute \(T(\vec e_1)\text{,}\) \(T(\vec e_2)\text{,}\) and \(T(\vec e_3)\text{.}\)
(b)
Find the standard matrix for \(T\text{.}\)
Fact 20.27.
Because every linear map \(T:\IR^n\to\IR^m\) has a linear combination of the variables in each component, and thus \(T(\vec e_i)\) yields exactly the coefficients of \(x_i\text{,}\) the standard matrix for \(T\) is simply an array of the coefficients of the \(x_i\text{:}\)
\begin{equation*}
T\left(\left[\begin{array}{c}x\\y\\z\\w\end{array}\right]\right)
=
\left[\begin{array}{c}
ax+by+cz+dw \\
ex+fy+gz+hw
\end{array}\right]
\hspace{2em}
A
=
\left[\begin{array}{cccc}
a & b & c & d \\
e & f & g & h
\end{array}\right]
\end{equation*}
Since the formula for a linear transformation \(T\) and its standard matrix \(A\) may both be used to compute the transformation of a vector \(\vec x\text{,}\) we will often write \(T(\vec x)\) and \(A\vec x\) interchangeably:
\begin{equation*}
T\left(\left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\right]\right)
=
\left[\begin{array}{c}
ax_1+bx_2+cx_3+dx_4 \\
ex_1+fx_2+gx_3+hx_4
\end{array}\right]
=
A\left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\right]
=
\left[\begin{array}{cccc}
a & b & c & d \\
e & f & g & h
\end{array}\right]
\left[\begin{array}{c}x_1\\x_2\\x_3\\x_4\end{array}\right]
\end{equation*}
Activity 20.28.
(a)
Explain and demonstrate how to compute the standard matrix for the linear transformation \(S:\mathbb{R}^2 \to \mathbb{R}^4\) given by
\begin{equation*}
S\left( \left[\begin{array}{c} x_{1} \\ x_{2} \end{array}\right] \right) = \left[\begin{array}{c} 9 \, x_{1} - 2 \, x_{2} \\ -3 \, x_{1} \\ 5 \, x_{1} - x_{2} \\ -6 \, x_{2} \end{array}\right]
\end{equation*}
by computing transformations of the standard basic vectors:
\begin{equation*}
S(\vec e_1)=\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\hspace{1em}S(\vec e_2)=\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right]\hspace{1em}\rightarrow\hspace{1em}\left[\begin{array}{cc} \unknown & \unknown \\ \unknown & \unknown \\ \unknown & \unknown \\\unknown & \unknown \end{array}\right]
\end{equation*}
Answer.
\begin{equation*}
\left[\begin{array}{cc} 9 & -2 \\ -3 & 0 \\ 5 & -1 \\ 0 & -6 \end{array}\right]
\end{equation*}
(b)
Let \(T:\mathbb{R}^4 \to \mathbb{R}^3\) be the linear transformation given by the standard matrix
\begin{equation*}
\left[\begin{array}{cccc} -2 & -4 & 2 & -2 \\ -4 & 3 & -3 & 2 \\ 5 & 0 & 2 & -6 \end{array}\right].
\end{equation*}
Explain and demonstrate how to compute \(T\left(\left[\begin{array}{c} -5 \\ 0 \\ -3 \\ -2 \end{array}\right]\right)\) by using the values of transformed standard basic vectors:
\begin{equation*}
T\left(\left[\begin{array}{c} -5 \\ 0 \\ -3 \\ -2 \end{array}\right]\right)=\unknown T(\vec e_1)+\unknown T(\vec e_2)+\unknown T(\vec e_3)+\unknown T(\vec e_4)
\end{equation*}
Answer.
\begin{equation*}
T\left(\left[\begin{array}{c} -5 \\ 0 \\ -3 \\ -2 \end{array}\right]\right)=\left[\begin{array}{c} 8 \\ 25 \\ -19 \end{array}\right]
\end{equation*}
Subsection 20.2.3 Individual Practice
Activity 20.29.
Consider the linear transformation \(R\colon\IR^2\to\IR^2\) given by rotating vectors about the origin through an angle of \(\frac{\pi}{4}=45^\circ\text{.}\)
(a)
If \(\vec{e}_1,\vec{e}_2\) are the standard basis vectors of \(\IR^2\text{,}\) calculate \(R(\vec{e}_1),R(\vec{e}_2)\text{.}\)
(b)
What is the standard matrix representing \(R\text{?}\)
Activity 20.30.
Consider the linear transformation \(S\colon\IR^2\to\IR^2\) given by reflecting vectors across the line \(x_1=x_2\text{.}\)
(a)
If \(\vec{e}_1,\vec{e}_2\) are the standard basis vectors of \(\IR^2\text{,}\) calculate \(S(\vec{e}_1),S(\vec{e}_2)\text{.}\)
(b)
What is the standard matrix representing \(S\text{?}\)
Subsection 20.2.4 Videos
Subsection 20.2.5 Exercises
Exercises available at
https://tbil.org/preview/linear-algebra/exercises/#/bank/AT2/.Subsection 20.2.6 Mathematical Writing Explorations
We can represent images in the plane \(\mathbb{R}^2\) using vectors, and manipulate those images with linear transformations. We introduce some notation in these explorations that is needed for their completion, but is not essential to the rest of the text. These have a geometric flair to them, and can be understood by thinking of geometric transformations in terms of standard matrices.
Given two vectors \(\vec{v} = \left[\begin{array}{c}v_1\\v_2\\ \vdots \\ v_n\end{array}\right]\) and \(\vec{w} = \left[\begin{array}{c}w_1 \\ w_2\\ \vdots \\ w_n\end{array}\right]\text{,}\) we define the dot product as
\begin{equation*}
\vec{v}\cdot \vec{w} = v_1w_1 + v_2w_2 + \cdots v_nw_n.
\end{equation*}
Exploration 20.31.
For each of the following properties, determine if it is held by the dot product. Either provide a proof that the property holds, or provide a counter-example if it does not.
- Distributive over addition (e.g., (\(\vec{u} + \vec{v})\cdot \vec{w} = \vec{u}\cdot\vec{w} + \vec{v}\cdot\vec{w})?\)
- Associative?
- Commutative?
Exploration 20.32.
Given the properties you proved in the last exploration, could the dot product take the place of \(\oplus\) as a vector space operation on \(\mathbb{R}^n\text{?}\)
Exploration 20.33.
Is the dot product a linear operator? That is, given vectors \(\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^n\text{,}\) and \(k,m \in \mathbb{R}\text{,}\) is it true that
\begin{equation*}
\vec{u} \cdot (k\vec{v} + m\vec{w}) = k(\vec{u} \cdot \vec{v}) + m(\vec{u}\cdot\vec{w}).
\end{equation*}
Prove or provide a counter-example.
Exploration 20.34.
Assume \(\vec{v} = \left[\begin{array}{c}v_1\\v_2\\ \vdots \\ v_n\end{array}\right]\) and define the length of a vector by
\begin{equation*}
|\vec{v}| = \left(\sum_{i=1}^n v_i^2 \right)^{1/2}\text{.}
\end{equation*}
Prove that \(|\vec{u}| = |\vec{v}|\) if an only if \(\vec{u} + \vec{v}\) and \(\vec{u} - \vec{v}\) are perpendicular. You may use the fact (try and prove it!) that two vectors are perpendicular if and only if their dot product is zero.
Exploration 20.35.
- A dilation is given by by mapping a vector \(\vec{v} = \left[\begin{array}{c}x\\y\end{array}\right]\) to some scalar multiple of \(\vec{v}\text{.}\)
- A rotation is given by \(\vec{v} \mapsto \left[\begin{array}{c} \cos(\theta)x - \sin(\theta)y\\ \cos(\theta)y + \sin(\theta)x\end{array}\right].\)
- A reflection of \(\vec{v}\) over a line \(l\) can be found by first finding a vector \(\vec{l} = \left[\begin{array}{c} l_x\\l_y\end{array}\right]\) along \(l\text{,}\) then \(\vec{v} \mapsto 2\frac{\vec{l}\cdot\vec{v}}{\vec{l}\cdot\vec{l}}\vec{l} - \vec{v}.\)
Represent each of the following transformations with respect to the standard basis in \(\mathbb{R}^2\text{.}\)
- Rotation through an angle \(\theta\text{.}\)
- Reflection over a line \(l\) passing through the origin.
- Dilation by some scalar \(s\text{.}\)
Prove that each transformation is linear, and that your matrix representations are correct.
Subsection 20.2.7 Sample Problem and Solution
Sample problem Example C.13.
