Use MATLAB to generate a matrix W and a vector x by setting

The columns of W can be used to form an ordered basis:

Let $L:{ℝ}^5\to {ℝ}^5$ be a linear operator such that

and

1. Determine the matrix A representing L with respect to F, and enter it in MATLAB.

2. Use MATLAB to compute the coordinate vector $\mathbf{y}={\text{W}}^{-1}\mathbf{x}$ of x with respect to F.

3. Use A to compute the coordinate vector z of L (x) with respect to F.

4. W is the transition matrix from F to the standard basis for ${ℝ}^5$. Use W to compute the coordinate vector of L (x) with respect to the standard basis.

Set $\text{A}=\mathsf{\text{t}}\mathsf{\text{r}}\mathsf{\text{i}}\mathsf{\text{u}}\mathsf{\left(}\mathsf{\text{o}}\mathsf{\text{n}}\mathsf{\text{e}}\mathsf{\text{s}}\mathsf{\left(}\mathsf{5}\mathsf{\right)}\mathsf{\right)}\mathsf{*}\mathsf{\text{t}}\mathsf{\text{r}}\mathsf{\text{i}}\mathsf{\text{l}}\mathsf{\left(}\mathsf{\text{o}}\mathsf{\text{n}}\mathsf{\text{e}}\mathsf{\text{s}}\mathsf{\left(}\mathsf{5}\mathsf{\right)}\mathsf{\right)}$. If L denotes the linear operator defined by $\text{L}\left(\mathbf{\text{x}}\right)=\text{A}\mathbf{\text{x}}$ for all x in ${ℝ}^n$, then A is the matrix representing L with respect to the standard basis for R⁵. Construct a $5\times 5$ matrix U by setting

Use the MATLAB function rank to verify that the column vectors of U are linearly independent. Thus, $E=\left\{{\mathbf{u}}_1,{\mathbf{u}}_2,{\mathbf{u}}_3,{\mathbf{u}}_4,{\mathbf{u}}_5\right\}$ is an ordered basis for ${ℝ}^5$. The matrix U is the transition matrix from E to the standard basis.

1. Use MATLAB to compute the matrix B representing L with respect to E. (The matrix B should be computed in terms of A, U, and $U^{-1}$).

2. Generate another matrix by setting

   $$V=\mathsf{\text{t}}\mathsf{\text{o}}\mathsf{\text{e}}\mathsf{\text{p}}\mathsf{\text{l}}\mathsf{\text{i}}\mathsf{\text{t}}\mathsf{\text{z}}\left(\left[1,0,1,1,1\right]\right)$$

   Use MATLAB to check that V is nonsingular. It follows that the column vectors of V are linearly independent and hence form an ordered basis F for ${ℝ}^5$. Use MATLAB to compute the matrix C, which represents L with respect to F. (The matrix C should be computed in terms of A, V, and $V^{-1}$.)

3. The matrices B and C from parts (a) and (b) should be similar. Why? Explain. Use MATLAB to compute the transition matrix S from F to E. Compute the matrix C in terms of B, S, and $S^{-1}$. Compare your result with the result from part (b).

Let

Let $L:{ℝ}^n\to {ℝ}^n$ be a linear transformation. If $L\left({\mathbf{x}}_1\right)=L\left({\mathbf{x}}_2\right)$, then the vectors ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$ must be equal.

If $L_1$ and $L_2$ are both linear operators on a vector space V, then $L_1+L_2$ is also a linear operator on V, where $L_1+L_2$ is the mapping defined by

$\left(L_1+L_2\right)\left(\text{v}\right)=L_1\left(\text{v}\right)+L_2\left(\text{v}\right)$ for all $\mathbf{v}\in V$.

If $L:V\to V$ is a linear transformation and $\mathbf{x}\in \text{ker}\left(L\right)$, then $L(\mathbf{v}+\mathbf{x})=L\left(\mathbf{v}\right)$ for all $\text{v}\in V$.

If $L_1$ rotates each vector x in ${ℝ}^2$ by 60° and then reflects the resulting vector about the x-axis, and if $L_2$ is a transformation that does the same two operations, but in the reverse order, then $L_1=L_2$.

The set of all vectors x used in the homogeneous coordinate system (see the application on computer graphics and animation in Section 4.2) forms a subspace of ${ℝ}^3$.

and set $B=S^{-1}*A*S$. The matrices A and B are similar. Use MATLAB to verify that the following properties hold for these two matrices:

1. $\det \left(B\right)=\det \left(A\right)$

2. $B^T=S^T{A}^T{\left(S^T\right)}^{-1}$

3. $B^{-1}=S^{-1}{A}^{-1}S$

4. $B^9=S^{-1}{A}^{9}S$

5. $B-3I=S^{-1}\left(A-3I\right)S$

6. $\det \left(B-3I\right)=\det \left(A-3I\right)$

7. $\text{tr}\left(B\right)=\text{tr}\left(A\right)$(Note that the trace of a matrix A can be computed with the MATLAB command trace.)

   These properties will hold in general for any pair of similar matrices (see Exercises 11-15 of Section 4.3).

Let $L:{ℝ}^2\to {ℝ}^2$ be a linear transformation, and let A be the standard matrix representation of L. If $L^2$ is defined by

then $L^2$ is a linear transformation and its standard matrix representation is $A^2$.

Let $E=\left\{{\mathbf{x}}_1,{\mathbf{x}}_2,\mathrm{\dots },{\mathbf{x}}_n\right\}$ be an ordered basis for ${ℝ}^n$. If $L_1:{ℝ}^n\to {ℝ}^n$ and $L_2:{ℝ}^n\to {ℝ}^n$ have the same matrix representation with respect to E, then $L_1=L_2$.

Let $L:{ℝ}^n\to {ℝ}^n$ be a linear transformation. If A is the standard matrix representation of L, then an $n\times n$ matrix B will also be a matrix representation of L if and only if B is similar to A.

Let A, B, and C be $n\times n$ matrices. If A is similar to B and B is similar to C, then A is similar to C.

Any two matrices with the same trace are similar. [This statement is the converse of part (b) of Exercise 15 in Section 4.3.]

Determine whether the following are linear operators on ${ℝ}^2$:

1. L is the operator defined by $L\left(\mathbf{x}\right)={\left(x_1+x_2,x_1\right)}^T\mathrm{.}$

2. L is the operator defined by $L\left(\mathbf{x}\right)={\left(x_1{x}_2,x_1\right)}^T\mathrm{.}$

Let L be a linear operator on ${ℝ}^2$ and let

If

find the value of $L\left({\mathbf{v}}_3\right)$.

Let L be the linear operator on ${ℝ}^3$ defined by

and let $S=\text{Span}\left({\left(1,0,1\right)}^T\right)$.

1. Find the kernel of L.

2. Determine L (S).

Let L be the linear operator on ${ℝ}^3$ defined by

Determine the range of L.

Let $L:{ℝ}^2\to {ℝ}^3$ be defined by

Find a matrix A such that $L\left(\mathbf{x}\right)=A\mathbf{x}$ for each x in ${ℝ}^2$.

Let L be the linear operator on ${ℝ}^2$ that rotates a vector by 30°. in the counterclockwise direction and then reflects the resulting vector about the y-axis. Find the standard matrix representation of L.

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

Find the matrix representation of L with respect to the homogeneous coordinate system.

Let

and let L be the linear operator that rotates vectors in ${ℝ}^2$ by 45°. in the counterclockwise direction. Find the matrix representation of L with respect to the ordered basis $\left[{\mathbf{u}}_1,{\mathbf{u}}_2\right]$.

Let

and

and let L be a linear operator on ${ℝ}^2$ whose matrix representation with respect to the ordered basis is $\left\{{\mathbf{u}}_1,{\mathbf{u}}_2\right\}$ is

1. Determine the transition matrix from the basis $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ to the basis $\left\{{\mathbf{u}}_1,{\mathbf{u}}_2\right\}$.

2. Find the matrix representation of L with respect to $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$.

Let A and B be similar matrices.

1. Show that $\det \left(A\right)=\det \left(B\right)$.

2. Show that if λ is any scalar, then $\det \left(A-\lambda I\right)=\det \left(B-\lambda I\right)$.