Given the bases \(\mathcal{B}=\{b_1,b_2\}\) and \(b_1=\begin{bmatrix}1\\2\end{bmatrix}\), \(b_2=\begin{bmatrix}3\\4\end{bmatrix}\) find the coordinates of \(\begin{bmatrix}0\\2\end{bmatrix}\).

Given the bases \(\mathcal{B}=\{b_1,b_2\}\) and \(\mathcal{D}=\{d_1,d_2\}\), which matrix below is the change of coordinates matrix from \(\mathcal{D}\) to \(\mathcal{B}\) if \(b_1=\begin{bmatrix}1\\2\end{bmatrix}\), \(b_2=\begin{bmatrix}3\\4\end{bmatrix}\), \(d_1=\begin{bmatrix}1\\4\end{bmatrix}\), and \(d_2=\begin{bmatrix}0\\8\end{bmatrix}\)

Let \(A\) be a \(3\times5\) matrix, \(B\) be a \(3\times 2\) matrix, and \(C\) be a \(3\times 5\) matrix.  Which of the following is a \(5\times 5\) matrix:

What is the 2nd row of the inverse of the matrix \(A=\begin{bmatrix} 2 & 1 & -1 \\ 1 & 0 & -3 \\ 6 & 4 & -1 \end{bmatrix}\)?

Which of the following are linear transformations? I) \( T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+3y\\0\end{bmatrix}\)  II) \( T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}x+y+1\\2x-y\end{bmatrix}\)    III) \(T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}xy\\2x-y\end{bmatrix}\)

Which of the following sets is a basis for the null space of the matrix \(A\) whose reduced row echelon form is \(\begin{bmatrix} 1 &  2 & 0 & 1 & 0\\0 &  0 & 1 & -1 & 0 \\ 0 &  0 & 0 & 0 & 1 \end{bmatrix}?\)

Let \(T\colon \mathbb{R}^{2}\to \mathbb{R}^{2}\) denote the linear transformation given by counterclockwise rotation about the origin by \(3\pi/4\) (radians).Let \(A\) be the matrix of \(T\) with respect to the standard basis of \(\mathbb{R}^{2}\).  Which of the following matrices is equal to \(A^{2}\)?

Use the two matrices:\(A\) = \(\begin{bmatrix} 3 & 2 &-1 &16 & 1 \\ 2 & 2 & -4 & 12 &1 \\ 1 & -1 & 8 & 2 & 0 \\ 1 & 1 & -2 & 6 &1\end{bmatrix}\), \(B\) = \(\begin{bmatrix}1& 0 &3 &4 &0 \\ 0 & 1 & -5 & 2 & 0 \\0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0\end{bmatrix}\) The matrix \(B\) is the reduced row echelon form of \(A\) (you do not have to check this.) Which set of vectors below is a basis for the column space of \(A\)?

Consider the linear system \(\begin{bmatrix} 2 & -3 \\ -6 & 9 \\ 4 & -7\end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} 1 \\ h \\ k\end{bmatrix} \) where \(h\) and \(k\) are real numbers. Which one of the following statements is true  about the solution?

How many absences in the course are equal to two weeks of absences?  Note: Having two weeks of absences means you are over the limit on absences allowed; you will be dropped from the course. .