Determine the dimensions of Nul A and Col A for the matrices shown.

$$A=\left[ \begin{array}{rrrrrr}{1} & {3} & {-4} & {2} & {-1} & {6} \\ {0} & {0} & {1} & {-3} & {7} & {0} \\ {0} & {0} & {0} & {1} & {4} & {-3} \\ {0} & {0} & {0} & {0} & {0} & {0}\end{array}\right]$$

Use the given information about f(x) and its derivatives to determine whether Trapezoidal Rule would be exact, underestimate or overestimate the integral (or there’s not enough information to tell). $f^{\prime \prime}(x)>0, f^{\prime}(x)<0$

What are the differences and similarities between the quantization of angular momentum in the Bohr model and the Schrodinger theory?

Show that the space $C(\mathbb{R})$ of all continuous functions defined on the real line is an infinite-dimensional space.

The first four Laguerre polynomials are 1, $1-t, 2-4 t+t^{2}$, and $6-18 t+9 t^{2}-t^{3}$. Show that these polynomials form a basis of $\mathbb{P}_3$.

Determine if the statement is true or false, and justify your answer. (a) If A and B are equivalent matrices, then col(A) = col(B). (b) If Ax = bi s a consistent linear system, then bi s in row(A).

Assume that the matrix A is row equivalent to B. Without calculations, list rank A and dim Nul A. Then find bases for Col A, Row A, and Nul A.

$$A=\left[\begin{array}{rrrrr} 1 & 3 & 4 & -1 & 2 \\ 2 & 6 & 6 & 0 & -3 \\ 3 & 9 & 3 & 6 & -3 \\ 3 & 9 & 0 & 9 & 0 \end{array}\right]$$

,

$$B=\left[\begin{array}{rrrrr} 1 & 3 & 4 & -1 & 2 \\ 0 & 0 & 1 & -1 & 1 \\ 0 & 0 & 0 & 0 & -5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$

Justify the following equality: $\operatorname{dim} \operatorname{Row} A+$ nullity $A=n$, the number of columns of $A$

Determine the dimensions of Nul A and Col A for the matrices.

$$A=\left[\begin{array}{rrrrrr}1 & 3 & -4 & 2 & -1 & 6 \\ 0 & 0 & 1 & -3 & 7 & 0 \\ 0 & 0 & 0 & 1 & 4 & -3 \\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$

In $\mathbb{P}_2$ find the change-of-coordinates matrix from the basis $\mathcal{B}=\left\{1-2 t+t^{2}, 3-5 t+4 t^{2}, 2 t+3 t^{2}\right\}$ to the standard basis $\mathcal{C}=\left\{1, t, t^{2}\right\}$. Then find the $\mathcal{B}$-coordinate vector for -1+2t.

Let $\mathcal{U}=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ and $\mathcal{W}=\left\{\mathbf{w}_{1}, \mathbf{w}_{2}\right\}$ be bases for V, and let P be a matrix whose columns are $\left[\mathbf{u}_{1}\right] w$ and $\left[\mathbf{u}_{2}\right] w$. Which of the following equations is satisfied by P for all x in V? (i) $[\mathbf{x}]_{u}=P[\mathbf{x}]_{w}$, (ii) $[\mathbf{x}]_{\mathcal{W}}=P[\mathbf{x}]_{\mathcal{U}}$

Find a nonzero signal in the range of $M_3$ from Exercise listed below.

Exercise:

Show that $M_3\left(\left\{x_k\right\}\right)=\left\{\frac{1}{3}\left(x_{k-2}+x_{k-1}+x_k\right)\right\}$ is a linear time invariant transformation.

Find a nonzero signal in the kernel of $M_3$ from Exercise listed below.

Exercise:

Show that $M_3\left(\left\{x_k\right\}\right)=\left\{\frac{1}{3}\left(x_{k-2}+x_{k-1}+x_k\right)\right\}$ is a linear time invariant transformation.

In the Exercise below, $\mathcal{B}$ and $\mathcal{C}$ are bases for a vector space $V$. Mark the statement True or False (T/F). Justify the answer.

(T/F) The columns of$\underset{\mathcal{C} \leftarrow \mathcal{B}}{P}$ are linearly independent.