Review Problems for the Final Exam

Suppose \(A\) is a \(3\times 5\) matrix, and \(\text {rank}(A)=2\).
(a)
Which of the following are possibilities for a non-zero vector \(\vec {b}\)? (Check all that apply.)
\(A\vec {x}=\vec {b}\) has a unique solution. \(A\vec {x}=\vec {b}\) has infinitely many solutions. \(A\vec {x}=\vec {b}\) has no solutions.
(b)
Find the following dimensions.
\[\text {dim}(\text {null}(A))=\answer {3},\quad \text {dim}(\text {col}(A))=\answer {2},\quad \text {dim}(\text {row}(A))=\answer {2}\]
(c)
Let \(T\) be a linear transformation induced by \(A\). What is true about \(T\)? (Check all that apply.)
\(T:\RR ^5\rightarrow \RR ^3\) \(T:\RR ^3\rightarrow \RR ^5\) \(\vec {0}\) is the ONLY vector that maps to \(\vec {0}\). Infinitely many vectors map to \(\vec {0}\). \(\text {dim}(\text {im}(T))=2\) \(\text {dim}(\text {im}(T))=3\) \(\text {dim}(\text {im}(T))=5\)
Suppose \(A\) is a \(7\times 2\) matrix, and \(\text {rank}(A)=2\).
(a)
Which of the following are possibilities for a non-zero vector \(\vec {b}\)? (Check all that apply.)
\(A\vec {x}=\vec {b}\) has a unique solution. \(A\vec {x}=\vec {b}\) has infinitely many solutions. \(A\vec {x}=\vec {b}\) has no solutions.
(b)
Which of the following is true about the columns of \(A\)?
Columns of \(A\) are linearly independent. Columns of \(A\) are linearly dependent. Not enough information is given to determine linear dependence/independence of the columns of \(A\).
(c)
Let \(T\) be a linear transformation induced by \(A\). What is true about \(T\)? (Check all that apply.)
\(T:\RR ^7\rightarrow \RR ^2\) \(T:\RR ^2\rightarrow \RR ^7\) \(\vec {0}\) is the ONLY vector that maps to \(\vec {0}\). Infinitely many vectors map to \(\vec {0}\). \(\text {dim}(\text {im}(T))=2\) \(\text {dim}(\text {im}(T))=7\)
You may use technology for this problem. Let
\[S=\text {span}\left (\begin{bmatrix}1\\-3\\14\end{bmatrix},\begin{bmatrix}17\\0\\16\end{bmatrix},\begin{bmatrix}10\\4\\-8\end{bmatrix}\right )\]
.
(a)
Describe \(S\).
\(S\) is ALL of \(\RR ^3\). \(S\) is a plane in \(\RR ^3\). \(S\) is a line in \(\RR ^3\).
(b)
Let \(\mathcal {B}=\left \{\begin{bmatrix}1\\-3\\14\end{bmatrix},\begin{bmatrix}17\\0\\16\end{bmatrix},\begin{bmatrix}10\\4\\-8\end{bmatrix}\right \}\). Is \(\mathcal {B}\) a basis for \(S\)?
Yes No
Let \(A\) be a \(3\times 3\) matrix. Suppose \(\text {rref}(A)=\begin{bmatrix}1 & 0 &-2\\0 & 1 & 1\\0 & 0& 0\end{bmatrix}\). What do you know (or can deduce) from this information? (Check all that apply.)
\(\det (A)=0\) I can find the row space of \(A\). I can find the column space of \(A\). Columns of \(A\) are linearly dependent. Rows of \(A\) are linearly dependent. \(A\) is not invertible. The null space of \(A\) has infinitely many elements. 0 is an eigenvalue of \(A\).
Find the standard matrix, \(A\), of the linear transformation \(T:\RR ^2\rightarrow \RR ^2\) if
\[T\left (\begin{bmatrix}1\\1\end{bmatrix}\right )=\begin{bmatrix}-1\\7\end{bmatrix},\quad T\left (\begin{bmatrix}0\\2\end{bmatrix}\right )=\begin{bmatrix}-4\\8\end{bmatrix}\]
\[A=\begin{bmatrix}\answer {1} & \answer {-2}\\\answer {3} & \answer {4}\end{bmatrix}\]
Find the determinant of \(A\) if
\[A=\begin{bmatrix}1 & 3 & 4 & 1\\k & 0 & 0 & 2\\0 & 1 & -2 & 1\\1 & -1 & 0 & 1\end{bmatrix}\]
where \(k\) is a constant.
\[\mbox {det}(A)=\answer {16k-24}\]

For what value of \(k\) is \(A\) singular?

\[k=\answer {1.5}\]
Use technology to find the line of best fit.

Enter your computed coefficients to three decimal places.

\[y=\answer [tolerance=0.01]{-0.915}x+\answer [tolerance=0.01]{0.393}\]
Plot your your line of best fit in the Desmos interactive above.

True or False? If False, you should come up with a counterexample.

Suppose \(A\) and \(B\) are \(n\times n\) matrices (\(n\geq 2\)).

(a)
\(\det {(AB)}=\det {A}\det {B}\)
True False
(b)
If \(\det {A}=\frac {1}{\det {B}}\), then \(A=B^{-1}\).
True False
(c)
\(\det {(A+B)}=\det {A}+\det {B}\).
True False
(d)
If \(A=2B\), then \(\det {A}=2\det {B}\).
True False
(e)
If \(A=-B\), then \(\det {A}=-\det {B}\) when \(n\) is odd, and \(\det {A}=\det {B}\) when \(n\) is even.
True False
Let \(B\) be a parallelepiped determined by vectors \(\vec {u}\), \(\vec {v}\), and \(\vec {w}\) in \(\RR ^3\). Suppose the volume of \(B\) is 10 cubic units. Let \(B'\) be a parallelepiped determined by \(-\vec {u}\), \(2\vec {v}\), and \(\vec {w}+\vec {u}\). Find the volume of \(B'\).
\[\text {volume of }B' = \answer {20}\,\text {cubic units}\]
Find eigenvalues of \(A=\begin{bmatrix} 2 & 4\\5 & 3\end{bmatrix}\).
\[\text {Eigenvalues (in increasing order): }\lambda _1=\answer {-2}, \quad \lambda _2=\answer {7}\]

Compute a basis for the eigenspace associated with each of these eigenvalues.

A basis for \(\mathcal {S}_{\lambda _1}\): \(\left \{\begin{bmatrix}1\\\answer {-1}\end{bmatrix}\right \}\)

A basis for \(\mathcal {S}_{\lambda _2}\): \(\left \{\begin{bmatrix}4\\\answer {5}\end{bmatrix}\right \}\)

Let \(A=\begin{bmatrix} 2 & 3 & -3 \\ 1 & 0 & -1 \\ 1 & 1 & -2 \end{bmatrix}\) and let \(B=\begin{bmatrix} 0 & 1 & 0 \\ 3 & 0 & 1 \\ 2 & 0 & 0 \end{bmatrix}\).

Each of these matrices has the same characteristic polynomial, \((2-\lambda )(1+\lambda )^2\). Which of the following is true? (Justify your answer.)

\(A\) is diagonalizable but \(B\) is not diagonalizable. \(A\) is not diagonalizable but \(B\) is diagonalizable. Both \(A\) and \(B\) are diagonalizable. Neither \(A\) nor \(B\) is diagonalizable.

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Problem 3.3.26.

For square matrices \(A\) and \(B\). True or False? If False, you should come up with a counterexample. If True, can you give a proof?

(a)
If \(\lambda \) is an eigenvalue of \(A\), then \(\lambda - c\) is an eigenvalue of \(A - cI\).
True False
(b)
If \(\lambda \) is an eigenvalue of \(A\), then \(2\lambda \) is an eigenvalue of \(2A\).
True False
If \(A = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 3 & 7 \\ 0 & 7 & 3 \end{bmatrix}\), find an orthogonal matrix \(Q\) and a diagonal matrix \(D\) such that \(D=Q^{-1}AQ\).
\[Q = \begin{bmatrix} 1 & \answer {0} & \answer {0} \\ \answer {0} & 1/\sqrt {2} & 1/\sqrt {2} \\ \answer {0} & 1/\sqrt {2} & \answer {-1/\sqrt {2}} \end{bmatrix}\]
\[D=\begin{bmatrix}\answer {5} & 0 &0\\0&\answer {10} & 0\\0 & 0 &\answer {-4}\end{bmatrix}\]
Verify that the columns of \(Q\) are orthonormal. Are the rows of \(Q\) orthonormal?
Each set \(V\) given below is a subset of \(\mathbb {M}_{2,2}\) with the usual matrix operations. Is \(V\) a subspace of \(\mathbb {M}_{2,2}\)?
(a)
\(V\) is the set of \(2 \times 2\) matrices with zero determinant.
\(V\) is a vector space. \(V\) is not a vector space.
(b)
\(V\) is the set of all \(2 \times 2\) matrices whose entries sum to 0.
\(V\) is a vector space. \(V\) is not a vector space.

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 6.1.2.

Describe the following subspace of \(\mathbb {M}_{3,3}\).
\[\text {span}\left \{\begin{bmatrix}1 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 1\end{bmatrix}, \begin{bmatrix}0 & 1 & 0\\1 & 0 & 0\\0 & 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 & 1\\0 & 0 & 0\\1 & 0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0\end{bmatrix}\right \}\]

The subspace consists of:

Matrices \(A\) of \(\mathbb {M}_{3,3}\) such that \(A^T=A\). All invertible matrices of \(\mathbb {M}_{3,3}\). Upper and lower triangular matrices of \(\mathbb {M}_{3,3}\). All of \(\mathbb {M}_{3,3}\).
Find the matrix of the linear transformation \(T:\mathbb {M}_{2,2}\rightarrow \mathbb {M}_{2,2}\), given by \(T(A)=A^T\), if the basis of the domain (\(\mathcal {B}\)) and the basis of the codomain (\(\mathcal {D}\)) are given by
\[\mathcal {B}=\mathcal {D}=\left \{\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix},\begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\1 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\0 & 1\end{bmatrix}\right \} \]
\[\begin{bmatrix}\answer {1} & \answer {0} & \answer {0} & \answer {0}\\\answer {0} & \answer {0} & \answer {1} & \answer {0}\\\answer {0} & \answer {1} & \answer {0} & \answer {0}\\\answer {0} & \answer {0} & \answer {0} & \answer {1}\end{bmatrix}\]

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 9.1.3 (b).

Find the coordinates of \(\vec {v}=ax^2+bx+c\) of \(\mathbb {P}^2\) with respect to the ordered basis \(\mathcal {B}=\left \{x^2, x+1, x+2\right \}\).
\[[\vec {v}]_{\mathcal {B}}=\begin{bmatrix}\answer {a}\\\answer {2b-c}\\\answer {c-b}\end{bmatrix}\]
[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 9.1.1 (b).
Suppose \(T:\mathbb {M}_{2,2}\rightarrow \RR \) is a linear transformation such that
\[T\left (\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}\right )=3\]
\[T\left (\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\right )=-1\]
\[T\left (\begin{bmatrix}1 & 0\\1 & 0\end{bmatrix}\right )=0\]
\[T\left (\begin{bmatrix}0 & 0\\0 & 1\end{bmatrix}\right )=0\]
Find \(T\left (\begin{bmatrix}a & b\\c & d\end{bmatrix}\right )\)
\[T\left (\begin{bmatrix}a & b\\c & d\end{bmatrix}\right )=\answer {3a+2b-3c}\]
[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 7.1.4 (d).
Suppose \(T:\mathbb {P}^2\rightarrow \mathbb {P}^3\) is a linear transformation such that
\[T(x^2)=x^3,\quad T(x+1)=0,\quad T(x-1)=x\]
Find \(T(x^2+x+1)\).
\[T(x^2+x+1)=\answer {x^3}\]
[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 7.1.4 (c).