Section3.2Standard Matrices (AT2)

Subsection3.2.1Class Activities

Remark3.2.1.

Recall that a linear map $$T:V\rightarrow W$$ satisfies

1. $$T(\vec{v}+\vec{w}) = T(\vec{v})+T(\vec{w})$$ for any $$\vec{v},\vec{w} \in V\text{.}$$

2. $$T(c\vec{v}) = cT(\vec{v})$$ 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.

Activity3.2.2.

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

1. $$\displaystyle \left[\begin{array}{c} 6 \\ 3\end{array}\right]$$

2. $$\displaystyle \left[\begin{array}{c} -9 \\ 6 \end{array}\right]$$

3. $$\displaystyle \left[\begin{array}{c} -4 \\ -2 \end{array}\right]$$

4. $$\displaystyle \left[\begin{array}{c} 6 \\ -4 \end{array}\right]$$

Activity3.2.3.

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

1. $$\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]$$

2. $$\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]$$

3. $$\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]$$

4. $$\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]$$

Activity3.2.4.

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 is $$T\left(\left[\begin{array}{c} -2 \\ 0 \\ -3 \end{array}\right]\right)\text{?}$$

1. $$\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]$$

2. $$\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]$$

3. $$\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]$$

4. $$\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]$$

Activity3.2.5.

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 $$T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)\text{?}$$

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

2. The value of $$T\left(\left[\begin{array}{c} 0\\1\\0\end{array}\right]\right)\text{.}$$

3. The value of $$T\left(\left[\begin{array}{c} 1\\1\\1\end{array}\right]\right)\text{.}$$

4. Any of the above.

Definition3.2.7.

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} 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*}

Activity3.2.8.

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 $$T\text{.}$$

Activity3.2.9.

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 $$T\text{.}$$

Activity3.2.11.

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

Activity3.2.12.

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

(a)
\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*}
(b)
\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*}
(c)
\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*}

Subsection3.2.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/AT2.slides.html.

Subsection3.2.5Mathematical 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*}

Exploration3.2.13.

For each of the following properties, determine if it is held by the dot product. Either provide a proof it 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?

Exploration3.2.14.

Given the properties you proved in the last exploration, could the dot product take the place of $$\oplus$$ as a vector space operation on $$\mathbb{R}^n\text{?}$$

Exploration3.2.15.

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.

Exploration3.2.16.

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.

Exploration3.2.17.

• 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.

Subsection3.2.6Sample Problem and Solution

Sample problem Example B.1.15.