Show that each of the following are linear operators on ${\mathrm{ℝ}}^2$. Describe geometrically what each linear transformation accomplishes.

1. $L\left(\mathbf{x}\right)={\left(-x_1,x_2\right)}^T$

2. $L\left(\mathbf{x}\right)=-\mathbf{x}$

3. $L\left(\mathbf{x}\right)={\left(x_2,x_1\right)}^T$

4. $L\left(\mathbf{x}\right)=\frac{1}{2}\mathbf{x}$

5. $L\left(\mathbf{x}\right)=x_2{\mathbf{\text{e}}}_2$

Let L be the linear operator on ${ℝ}^2$ defined by

Express $x_1,x_2$, and $L\left(\text{x}\right)$ in terms of polar coordinates. Describe geometrically the effect of the linear transformation.

Let a be a fixed nonzero vector in ${\mathrm{ℝ}}^2$. A mapping of the form

is called a translation. Show that a translation is not a linear operator. Illustrate geometrically the effect of a translation.

Let $L:{\mathrm{ℝ}}^2\to {\mathrm{ℝ}}^2$ be a linear operator. If

and

find the value of $L({(\mathrm{7,\; 5})}^T)$.

Determine whether the following are linear transformations from ${\mathrm{ℝ}}^3$ into ${\mathrm{ℝ}}^2$:

1. $L\left(\mathbf{x}\right)={\left(x_2,x_3\right)}^T$

2. $L\left(\mathbf{x}\right)={\left(0,0\right)}^T$

3. $L\left(\mathbf{x}\right)={\left(1+x_1,x_2\right)}^T$

4. $L\left(\mathbf{x}\right)={\left(x_3,x_1+x_2\right)}^T$

Determine whether the following are linear transformations from ${\mathrm{ℝ}}^2$ into ${\mathrm{ℝ}}^3$:

1. $L\left(\mathbf{x}\right)={\left(x_1,x_2,1\right)}^T$

2. $L\left(\mathbf{x}\right)={\left(x_1,x_2,x_1+2x_2\right)}^T$

3. $L\left(\mathbf{x}\right)={\left(x_1,0,0\right)}^T$

4. $L\left(\mathbf{x}\right)={\left(x_1,x_2,x_1^2+x_2^2\right)}^T$

Determine whether the following are linear operators on ${\mathrm{ℝ}}^{n\times n}$:

1. $L\left(A\right)=2A$

2. $L\left(A\right)=A^T$

3. $L\left(A\right)=A+I$

4. $L\left(A\right)=A-A^T$

Let C be a fixed $n\times n$ matrix. Determine whether the following are linear operators on ${\mathrm{ℝ}}^{n\times n}$:

1. $L\left(A\right)=CA+AC$

2. $L\left(A\right)=C^{2}A$

3. $L\left(A\right)=A^{2}C$

Determine whether the following are linear transformations from $P_2$ to $P_3$:

1. $L(p\left(x\right))=xp\left(x\right)$

2. $L\left(p\left(x\right)\right)=x^2+p\left(x\right)$

3. $L\left(p\left(x\right)\right)=p\left(x\right)+xp\left(x\right)+x^{2}p\prime \left(x\right)$

For each $f\in C\left[0,1\right]$, define $L\left(f\right)=F$, where

Show that L is a linear operator on $C\left[0,1\right]$ and then find $L\left(e^x\right)$ and $L\left(x^2\right)$.

Determine whether the following are linear transformations from $C\left[0,1\right]$ into ${\mathrm{ℝ}}^1$:

1. $L\left(f\right)=f\left(0\right)$

2. $L\left(f\right)=\left|f\left(0\right)\right|$

3. $L\left(f\right)=\left[f\left(0\right)+f\left(1\right)\right]/2$

4. $L\left(f\right)={\left\{{\int }_0^1{\left[f\left(x\right)\right]}^{2}dx\right\}}^{1/2}$

Use mathematical induction to prove that if L is a linear transformation from V to W, then

Let $\left\{{\mathbf{\text{v}}}_1,\mathrm{\dots },{\mathbf{\text{v}}}_n\right\}$ be a basis for a vector space V, and let $L_1$ and $L_2$ be two linear transformations mapping V into a vector space W. Show that if

for each $i=1,\mathrm{\dots },n$, then $L_1=L_2$ [i.e., show that $L_1\left(\mathbf{\text{v}}\right)=L_2\left(\mathbf{\text{v}}\right)$ for all $\mathbf{\text{v}}\in V$].

Let L be a linear operator on ${\mathrm{ℝ}}^1$ and let $a=L\left(1\right)$. Show that $L\left(x\right)=ax$ for all $x\in {\mathrm{ℝ}}^1$.

Let L be a linear operator on a vector space V. Define $L^n$, $n\ge 1$, recursively by

Show that $L^n$ is a linear operator on V for each $n\ge 1$.

Let $L_1:U\to V$ and $L_2:V\to W$ be linear transformations, and let $L=L_2\circ L_1$ be the mapping defined by

for each $\mathbf{u}\in U$. Show that L is a linear transformation mapping U into W.

Determine the kernel and range of each of the following linear operators on ${\mathrm{ℝ}}^3$:

1. $L\left(\mathbf{\text{x}}\right)={(x_3,x_2,x_1)}^T$

2. $L\left(\mathbf{x}\right)={\left(x_1,x_2,0\right)}^T$

3. $L\left(\mathbf{x}\right)={\left(x_1,x_1,x_1\right)}^T$

Let S be the subspace of ${\mathrm{ℝ}}^3$ spanned by ${\mathbf{\text{e}}}_1$ and ${\mathbf{\text{e}}}_2$. For each linear operator L in Exercise 17, find $L\left(S\right)$.

Find the kernel and range of each of the following linear operators on $P_3$:

1. $L\left(p\left(x\right)\right)=xp\prime \left(x\right)$

2. $L\left(p\left(x\right)\right)=p\left(x\right)-p\prime \left(x\right)$

3. $L\left(p\left(x\right)\right)=p\left(0\right)x-p\left(1\right)$

Let $L:V\to W$ be a linear transformation, and let T be a subspace of W. The inverse image of T, denoted $L^{-1}\left(T\right)$, is defined by

Show that $L^{-1}\left(T\right)$ is a subspace of V.

A linear transformation $L:V\to W$ is said to be one-to-one if $L\left({\mathbf{\text{v}}}_1\right)=L\left({\mathbf{\text{v}}}_2\right)$ implies that ${\mathbf{\text{v}}}_1={\mathbf{\text{v}}}_2$ (i.e., no two distinct vectors ${\mathbf{\text{v}}}_1$, ${\mathbf{\text{v}}}_2$ in V get mapped into the same vector $\mathbf{\text{w}}\in W$). Show that L is one-to-one if and only if $\text{ker}\left(L\right)=\left\{{\mathbf{0}}_V\right\}$.

A linear transformation $L:V\to W$ is said to map V onto W if $L\left(V\right)=W$. Show that the linear transformation L defined by

maps ${\mathrm{ℝ}}^3$ onto ${\mathrm{ℝ}}^3$.

Which of the operators defined in Exercise 17 are one-to-one? Which map ${\mathrm{ℝ}}^3$ onto ${\mathrm{ℝ}}^3$?

Let A be a $2\times 2$ matrix, and let $L_A$ be the linear operator defined by

Show that

1. $L_A$ maps ${\mathrm{ℝ}}^2$ onto the column space of A.

2. if A is nonsingular, then $L_A$ maps ${\mathrm{ℝ}}^2$ onto ${\mathrm{ℝ}}^2$.

Let D be the differentiation operator on $P_3$, and let

Show that

1. D maps $P_3$ on to the subspace $P_2$, but $D:P_3\to P_2$ is not one-to-one.

2. $D:S\to P_3$ is one-to-one but not onto.