## Section3.1Linear Transformations (AT1)

### Subsection3.1.1Class Activities

#### Definition3.1.1.

A linear transformation (also called a linear map) is a map between vector spaces that preserves the vector space operations. More precisely, if $$V$$ and $$W$$ are vector spaces, a map $$T:V\rightarrow W$$ is called a linear transformation if

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

2. $$T(c\vec{v}) = cT(\vec{v})$$ for any $$c \in \IR,$$ and $$\vec{v} \in V\text{.}$$

In other words, a map is linear when vector space operations can be applied before or after the transformation without affecting the result.

#### Definition3.1.2.

Given a linear transformation $$T:V\to W\text{,}$$ $$V$$ is called the domain of $$T$$ and $$W$$ is called the co-domain of $$T\text{.}$$ Figure 23. A linear transformation with a domain of $$\IR^3$$ and a co-domain of $$\IR^2$$

#### Example3.1.3.

Let $$T : \IR^3 \rightarrow \IR^2$$ be given by

\begin{equation*} T\left(\left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} x-z \\ 3y \end{array}\right]. \end{equation*}

To show that $$T$$ is a linear transformation, we must verify that $$T(\vec{v}+\vec{w}) = T(\vec{v})+T(\vec{w})$$ by computing

\begin{equation*} T\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] + \left[\begin{array}{c} u \\ v \\ w \end{array}\right] \right) = T\left( \left[\begin{array}{c} x+u \\ y+v \\ z+w \end{array}\right] \right) = \left[\begin{array}{c} (x+u)-(z+w) \\ 3(y+v) \end{array}\right] \end{equation*}

and

\begin{equation*} T\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) + T\left( \left[\begin{array}{c} u \\ v \\ w \end{array}\right] \right) = \left[\begin{array}{c} x-z \\ 3y \end{array}\right] + \left[\begin{array}{c} u-w \\ 3v \end{array}\right]= \left[\begin{array}{c} (x+u)-(z+w) \\ 3(y+v) \end{array}\right]\text{,} \end{equation*}

and we must verify that $$T(c\vec{v}) = cT(\vec{v})$$ by computing

\begin{equation*} T\left(c\left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = T\left(\left[\begin{array}{c} cx \\ cy \\ cz \end{array}\right] \right) = \left[\begin{array}{c} cx-cz \\ 3cy \end{array}\right] \text{ and } cT\left(\left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = c\left[\begin{array}{c} x-z \\ 3y \end{array}\right] = \left[\begin{array}{c} cx-cz \\ 3cy \end{array}\right]\text{.} \end{equation*}

Therefore $$T$$ is a linear transformation.

#### Example3.1.4.

Let $$S : \IR^2 \rightarrow \IR^4$$ be given by

\begin{equation*} S\left(\left[\begin{array}{c} x \\ y \end{array}\right] \right) = \left[\begin{array}{c} x+y \\ x^2 \\ y+3 \\ y-2^x \end{array}\right] \end{equation*}

To show that $$S$$ is not linear, we only need to find one counterexample.

\begin{equation*} S\left( \left[\begin{array}{c} 0 \\ 1 \end{array}\right] + \left[\begin{array}{c} 2 \\ 3 \end{array}\right] \right) = S\left( \left[\begin{array}{c} 2 \\ 4 \end{array}\right] \right) = \left[\begin{array}{c} 6 \\ 4 \\ 7 \\ 0 \end{array}\right] \end{equation*}
\begin{equation*} S\left( \left[\begin{array}{c} 0 \\ 1 \end{array}\right] \right) + S\left( \left[\begin{array}{c} 2 \\ 3\end{array}\right] \right) = \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ 0 \end{array}\right] + \left[\begin{array}{c} 5 \\ 4 \\ 6 \\ -1 \end{array}\right] = \left[\begin{array}{c} 6 \\ 4 \\ 10 \\ -1 \end{array}\right] \end{equation*}

Since the resulting vectors are different, $$S$$ is not a linear transformation.

#### Activity3.1.6.

Let $$D:\P\to\P$$ be the derivative map defined by $$D(f(x))=f'(x)$$ for each polynomial $$f \in \P\text{.}$$ We recall from calculus that

\begin{equation*} D(f(x)+g(x))=f'(x)+g'(x)\text{,} \end{equation*}
and
\begin{equation*} D(cf(x))=cf'(x)\text{.} \end{equation*}

Which of the following can we conclude from these calculus rules?

1. $$\P$$ is not a vector space

2. $$D$$ is a linear map

3. $$D$$ is not a linear map

#### Activity3.1.7.

Let the polynomial maps $$S: \P_4 \rightarrow \P_3$$ and $$T: \P_4 \rightarrow \P_3$$ be defined by

\begin{equation*} S(f(x)) = 2f'(x)-f''(x) \hspace{3em} T(f(x)) = f'(x)+x^3\text{.} \end{equation*}

Compute $$S(x^4+x)\text{,}$$ $$S(x^4)+S(x)\text{,}$$ $$T(x^4+x)\text{,}$$ and $$T(x^4)+T(x)\text{.}$$ Based on these computations, can you conclude that either $$S$$ or $$T$$ is definitely not a linear transformation?

#### Observation3.1.9.

Showing $$T:V\to W$$ is not a linear transformation can be done by finding an example for any one of the following.

• Show $$T(\vec z)\not=\vec z$$ (where $$\vec z$$ is the additive identity of $$V$$ and $$W$$).

• Find $$\vec v,\vec w\in V$$ such that $$T(\vec v+\vec w)\not=T(\vec v)+T(\vec w)\text{.}$$

• Find $$\vec v\in V$$ and $$c\in \IR$$ such that $$T(c\vec v)\not=cT(\vec v)\text{.}$$

Otherwise, $$T$$ can be shown to be linear by proving the following in general.

• For all $$\vec v,\vec w\in V\text{,}$$ $$T(\vec v+\vec w)=T(\vec v)+T(\vec w)\text{.}$$

• For all $$\vec v\in V$$ and $$c\in \IR\text{,}$$ $$T(c\vec v)=cT(\vec v)\text{.}$$

Note the similarities between this process and showing that a subset of a vector space is or is not a subspace.

#### Activity3.1.10.

Continue to consider $$S: \P_4 \rightarrow \P_3$$ defined by

\begin{equation*} S(f(x)) = 2f'(x)-f''(x)\text{.} \end{equation*}
##### (a)

Verify that

\begin{equation*} S(f(x)+g(x))=2f'(x)+2g'(x)-f''(x)-g''(x) \end{equation*}

is equal to $$S(f(x))+S(g(x))$$ for all polynomials $$f,g \in \P_4\text{.}$$

##### (b)

Verify that $$S(cf(x))$$ is equal to $$cS(f(x))$$ for all real numbers $$c$$ and polynomials $$f \in \P_4\text{.}$$

##### (c)

Is $$S$$ linear?

#### Activity3.1.11.

Let polynomial maps $$S: \P \rightarrow \P$$ and $$T: \P \rightarrow \P$$ be defined by

\begin{equation*} S(f(x)) = (f(x))^2 \hspace{3em} T(f(x)) = 3xf(x^2) \end{equation*}
##### (a)

Note that $$S(0)=0$$ and $$T(0)=0\text{.}$$ So instead, show that $$S(x+1)\not= S(x)+S(1)$$ to verify that $$S$$ is not linear.

##### (b)

Prove that $$T$$ is linear by verifying that $$T(f(x)+g(x))=T(f(x))+T(g(x))$$ and $$T(cf(x))=cT(f(x))\text{.}$$

### Subsection3.1.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/AT1.slides.html.

### Subsection3.1.5Mathematical Writing Explorations

#### Exploration3.1.12.

If $$V,W$$ are vectors spaces, with associated zero vectors $$\vec{0}_V$$ and $$\vec{0}_W\text{,}$$ and $$T:V \rightarrow W$$ is a linear transformation, does $$T(\vec{0}_V) = \vec{0}_W\text{?}$$ Prove this is true, or find a counterexample.

#### Exploration3.1.13.

Assume $$f: V \rightarrow W$$ is a linear transformation between vector spaces. Let $$\vec{v} \in V$$ with additive inverse $$\vec{v}^{-1}\text{.}$$ Prove that $$f(\vec{v}^{-1}) = [f(\vec{v})]^{-1}\text{.}$$

### Subsection3.1.6Sample Problem and Solution

Sample problem Example B.1.14.