Standard Matrices (A2)

Section 3.2 Standard Matrices (A2)

Remark 3.2.1.

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 3.2.1.

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 \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right) = \left[\begin{array}{c} -3 \\ 2 \end{array}\right] \text{.}\) Compute \(T\left(\left[\begin{array}{c} 3 \\ 0 \\ 0 \end{array}\right]\right)\text{.}\)

\(\displaystyle \left[\begin{array}{c} 6 \\ 3\end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} -9 \\ 6 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} -4 \\ -2 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} 6 \\ -4 \end{array}\right]\)

Activity 3.2.2.

\(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)

Activity 3.2.3.

\(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)

Activity 3.2.4.

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 \(T\left(\left[\begin{array}{c} 0\\-4\\0\end{array}\right]\right)\text{.}\)
The value of \(T\left(\left[\begin{array}{c} 0\\1\\0\end{array}\right]\right)\text{.}\)
The value of \(T\left(\left[\begin{array}{c} 1\\1\\1\end{array}\right]\right)\text{.}\)
Any of the above.

Fact 3.2.2.

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, \(x_1\vec b_1+\dots+ x_n\vec b_n\text{,}\) 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 determine the transformation \(T\text{.}\)

Definition 3.2.3.

Since linear transformation \(T:\IR^n\to\IR^m\) is determined by the standard basis \(\{\vec e_1,\dots,\vec e_n\}\text{,}\) it's convenient to store this information in the \(m\times n\) standard matrix \([T(\vec e_1) \,\cdots\, T(\vec e_n)]\text{.}\)

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} 3 \\ 2\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} 5 \\ 0\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}3 & -1 & 5 \\ 2 & 4 & 0 \end{array}\right] . \end{equation*}

Activity 3.2.5.

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 3.2.6.

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 3.2.4.

Because every linear map \(T:\IR^m\to\IR^n\) 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 ordered list 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*}

Activity 3.2.7.

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix

\begin{equation*} \left[\begin{array}{ccc} 3 & -2 & -1 \\ 4 & 5 & 2 \\ 0 & -2 & 1 \end{array}\right] . \end{equation*}

(a)

Compute \(T\left(\left[\begin{array}{c} 1\\ 2 \\ 3 \end{array}\right] \right) \text{.}\)

(b)

Compute \(T\left(\left[\begin{array}{c} x\\ y \\ z \end{array}\right] \right) \text{.}\)

Activity 3.2.8.

Compute the following linear transformations of vectors given their standard matrices.

\begin{equation*} T_1\left(\left[\begin{array}{c}1\\2\end{array}\right]\right) \text{ for the standard matrix } A_1=\left[\begin{array}{cc}4&3\\0&-1\\1&1\\3&0\end{array}\right] \end{equation*}

\begin{equation*} T_2\left(\left[\begin{array}{c}1\\1\\0\\-3\end{array}\right]\right) \text{ for the standard matrix } A_2=\left[\begin{array}{cccc}4&3&0&-1\\1&1&3&0\end{array}\right] \end{equation*}

\begin{equation*} T_3\left(\left[\begin{array}{c}0\\-2\\0\end{array}\right]\right) \text{ for the standard matrix } A_3=\left[\begin{array}{ccc}4&3&0\\0&-1&3\\5&1&1\\3&0&0\end{array}\right] \end{equation*}

Exercises 3.2.1 Exercises

1.

Find 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 \\ y \end{array}\right] \right) = \left[\begin{array}{c} x - 5 \, y \\ -x + 6 \, y \\ 0 \\ x - 2 \, y \end{array}\right] . \end{equation*}
Let \(T:\mathbb{R}^ 2 \to \mathbb{R}^ 4 \) be the linear transformation given by the standard matrix
\begin{equation*} \left[\begin{array}{cc} -1 & 0 \\ -1 & -4 \\ 2 & 5 \\ -1 & 0 \end{array}\right] . \end{equation*}
Compute \(T\left( \left[\begin{array}{c} -2 \\ 8 \end{array}\right] \right)\text{.}\)

Answer

\begin{equation*} \left[\begin{array}{cc} 1 & -5 \\ -1 & 6 \\ 0 & 0 \\ 1 & -2 \end{array}\right] \end{equation*}
\begin{equation*} T\left( \left[\begin{array}{c} -2 \\ 8 \end{array}\right] \right)= \left[\begin{array}{c} 2 \\ -30 \\ 36 \\ 2 \end{array}\right] \end{equation*}

2.

Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) given by
\begin{equation*} S\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} -x + 3 \, y + 4 \, z \\ -x - z \\ 2 \, x - 2 \, y - z \\ -3 \, x + y - 6 \, z \end{array}\right] . \end{equation*}
Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 2 \) be the linear transformation given by the standard matrix
\begin{equation*} \left[\begin{array}{cccc} 2 & 6 & 1 & -3 \\ 1 & 3 & 1 & -2 \end{array}\right] . \end{equation*}
Compute \(T\left( \left[\begin{array}{c} -3 \\ -7 \\ 6 \\ -5 \end{array}\right] \right)\text{.}\)

Answer

\begin{equation*} \left[\begin{array}{ccc} -1 & 3 & 4 \\ -1 & 0 & -1 \\ 2 & -2 & -1 \\ -3 & 1 & -6 \end{array}\right] \end{equation*}
\begin{equation*} T\left( \left[\begin{array}{c} -3 \\ -7 \\ 6 \\ -5 \end{array}\right] \right)= \left[\begin{array}{c} -27 \\ -8 \end{array}\right] \end{equation*}

3.

Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 4 \to \mathbb{R}^ 2 \) given by
\begin{equation*} S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} -2 \, x + y - 2 \, {w} \\ x - y \end{array}\right] . \end{equation*}
Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 1 \) be the linear transformation given by the standard matrix
\begin{equation*} \left[\begin{array}{cccc} 1 & -1 & -2 & -1 \end{array}\right] . \end{equation*}
Compute \(T\left( \left[\begin{array}{c} 6 \\ -3 \\ 5 \\ -7 \end{array}\right] \right)\text{.}\)

Answer

\begin{equation*} \left[\begin{array}{cccc} -2 & 1 & 0 & -2 \\ 1 & -1 & 0 & 0 \end{array}\right] \end{equation*}
\begin{equation*} T\left( \left[\begin{array}{c} 6 \\ -3 \\ 5 \\ -7 \end{array}\right] \right)= \left[\begin{array}{c} 6 \end{array}\right] \end{equation*}

4.

Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 3 \) given by
\begin{equation*} S\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} x \\ y - 4 \, z \\ -y + 5 \, z \end{array}\right] . \end{equation*}
Let \(T:\mathbb{R}^ 2 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix
\begin{equation*} \left[\begin{array}{cc} -5 & 3 \\ 3 & -2 \\ 0 & 2 \end{array}\right] . \end{equation*}
Compute \(T\left( \left[\begin{array}{c} -5 \\ 3 \end{array}\right] \right)\text{.}\)

Answer

\begin{equation*} \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & -4 \\ 0 & -1 & 5 \end{array}\right] \end{equation*}
\begin{equation*} T\left( \left[\begin{array}{c} -5 \\ 3 \end{array}\right] \right)= \left[\begin{array}{c} 34 \\ -21 \\ 6 \end{array}\right] \end{equation*}

5.

Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 4 \to \mathbb{R}^ 2 \) given by
\begin{equation*} S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} 2 \, x - 3 \, y + z + 6 \, {w} \\ x - y + 2 \, {w} \end{array}\right] . \end{equation*}
Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix
\begin{equation*} \left[\begin{array}{cccc} 3 & 0 & 4 & 5 \\ 0 & 1 & 1 & 3 \\ -1 & -1 & -2 & -4 \end{array}\right] . \end{equation*}
Compute \(T\left( \left[\begin{array}{c} 5 \\ 8 \\ 7 \\ -2 \end{array}\right] \right)\text{.}\)

Answer

\begin{equation*} \left[\begin{array}{cccc} 2 & -3 & 1 & 6 \\ 1 & -1 & 0 & 2 \end{array}\right] \end{equation*}
\begin{equation*} T\left( \left[\begin{array}{c} 5 \\ 8 \\ 7 \\ -2 \end{array}\right] \right)= \left[\begin{array}{c} 33 \\ 9 \\ -19 \end{array}\right] \end{equation*}

Additional exercises available at checkit.clontz.org.