Chapter 6, Linear Maps Video Solutions, Linear Algebra

Problem 1

Define the map $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ by $T(x, y)=(x+y, x)$.
a. Show that $T$ is linear.
b. Show that $T$ is surjective.
c. Find $\operatorname{dim}(\operatorname{null}(T))$.
d. Find the matrix for $T$ with respect to the canonical basis of $\mathbb{R}^{2}$.
e. Find the matrix for $T$ with respect to the canonical basis for the domain $\mathbb{R}^{2}$ and the basis $((1,1),(1,-1))$ for the target space $\mathbb{R}^{2}$.
f. Show that the map $F: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by $F(x, y)=(x+y, x+1)$ is not linear.

Victor Salazar

Numerade Educator

15:53

Problem 2

Let $T \in \mathcal{L}\left(\mathbb{R}^{2}\right)$ be defined by
$$T\left(\begin{array}{l}x \\y\end{array}\right)=\left(\begin{array}{c}y \\-x\end{array}\right), \quad \text { for all }\left(\begin{array}{l}x \\y\end{array}\right) \in \mathbb{R}^{2}$$
a. Show that $T$ is surjective.
b. Find $\operatorname{dim}(\operatorname{null}(T))$.
c. Find the matrix for $T$ with respect to the canonical basis of $\mathbb{R}^{2}$.
d. Show that the map $F: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by $F(x, y)=(x+y, x+1)$ is not linear.

Millie Lopez

Numerade Educator

14:34

Problem 3

Consider the complex vector spaces $\mathbb{C}^{2}$ and $\mathbb{C}^{3}$ with their canonical bases, and define $S \in \mathcal{L}\left(\mathbb{C}^{3}, \mathbb{C}^{2}\right)$ be the linear map defined by $S(v)=A v, \forall v \in \mathbb{C}^{3}$, where $A$ is the matrix
$$A=M(S)=\left(\begin{array}{ccc}i & 1 & 1 \\2 i & -1 & -1\end{array}\right)$$
Find a basis for $\operatorname{null}(S)$.

Uma Kumari

Numerade Educator

04:54

Problem 4

Give an example of a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ having
the property that
$$\forall a \in \mathbb{R}, \forall v \in \mathbb{R}^{2}, f(a v)=a f(v)$$
but such that $f$ is not a linear map.

Anthony Ramos

Numerade Educator

02:08

Problem 5

Show that the linear map $T: \mathbb{F}^{4} \rightarrow \mathbb{F}^{2}$ is surjective if
$$\operatorname{null}(T)=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \in \mathbb{F}^{4} \mid x_{1}=5 x_{2}, x_{3}=7 x_{4}\right\}$$

Ashley Boni

Numerade Educator

Problem 6

Show that no linear map $T: \mathbb{F}^{5} \rightarrow \mathbb{F}^{2}$ can
have as its null space the set
$$\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \in \mathbb{F}^{5} \mid x_{1}=3 x_{2}, x_{3}=x_{4}=x_{5}\right\} .$$

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07:04

Problem 7

Describe the set of solutions $x=\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3}$ of the system of equations
$$\left.\begin{array}{rl}
x_{1}-x_{2}+x_{3} & =0 \\x_{1}+2 x_{2}+x_{3} & =0 \\
2 x_{1}+x_{2}+2 x_{3} & =0\end{array}\right\}$$

Jingyun Wang

Numerade Educator