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 3.2.2.
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 3.2.3.
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 \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right]\right)\text{.}\)
\(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)
Activity 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 \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right) = \left[\begin{array}{c} -3 \\ 2 \end{array}\right] \text{.}\) Compute \(T\left(\left[\begin{array}{c} -2 \\ 0 \\ -3 \end{array}\right]\right)\text{.}\)
\(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)
\(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)
Activity 3.2.5.
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 3.2.6.
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:
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 3.2.7.
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 \(\IR^3\text{.}\)
Then the standard matrix corresponding to \(T\) is
Activity 3.2.8.
Let \(T: \IR^4 \rightarrow \IR^3\) be the linear transformation given by
Write the standard matrix \([T(\vec e_1) \,\cdots\, T(\vec e_n)]\) for \(T\text{.}\)
Activity 3.2.9.
Let \(T: \IR^3 \rightarrow \IR^2\) be the linear transformation given by
(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 3.2.10.
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 \(x_i\text{:}\)
Activity 3.2.11.
Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix
(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 3.2.12.
Compute the following linear transformations of vectors given their standard matrices.
Subsection 3.2.1 Videos
Exercises 3.2.2 Exercises
Exercises available at checkit.clontz.org 1 .
https://checkit.clontz.org/#/banks/tbil-la/A2/