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\begin{center}
{\small \textbf{Bronx Community College of City University of New York}\\
\textbf{Department of Mathematics and Computer Science}}\\[6pt]
\fbox{MTH 42 Linear Algebra Review Sheet}
\end{center}

\vspace{10pt}



\begin{enumerate}[label=\textbf{\arabic*}.]

\item For the linear system $\left\{ 
\begin{array}{l} 
\phantom{2}x_1 + \phantom{2}x_2 + \phantom{2}x_3 = 1 \\ 
2x_1 + 2x_2 + \phantom{2}x_3 = 2 \\
\phantom{2}x_1 + \phantom{2}x_2 + 2x_3 = 3 \end{array} \right.$, answer the following:

\begin{enumerate}
\item Find matrices $\mathbf{A}$, $\mathbf{x}$ and $\mathbf{b}$ that express the linear system above as a matrix equation.
\item Find $\mathbf{A}^{-1}$ and express it as a product of elementary matrices.
\item Solve the system by multiplying $\mathbf{A}^{-1}$ and $\mathbf{b}$.
\end{enumerate}

\vspace{6pt}

\item Let $\left\{ 
\begin{array}{l} 
ax_1  + \phantom{2}\, b  x_2 = 2 \\ 
\phantom{a}x_1 + 2\, bx_2 = 1 
\end{array} 
\right.$. Find all values of $a, b$ so that the system has

\begin{enumerate}
\item exactly one solution; (b) no solution; (c) infinitely many solutions.
\end{enumerate}

\vspace{6pt}

\item Find $\mathbf{A}$ if $(5\mathbf{A}^T)^{-1} = \left[ \begin{array}{ll} 1 & 2 \\ 3 & 7 \end{array} \right]$.

\vspace{10pt}

\item Find values of $a, b$ and $c$ for which $\mathbf{A} = \left[ \begin{array}{ccc} 1 & a+b & c \\ 1 & 1 & a-b \\ 2 & 0 & 1 \end{array} \right]$ is symmetric.

\vspace{10pt}

\item Given $\left| \begin{array}{ll} a & b \\ c & d \end{array} \right| = n$, find the determinant of each matrix below.

\begin{equation*}
(a)\left[ \begin{array}{cc} a & c \\ b & d \end{array} \right] \quad
(b)\left[ \begin{array}{cc} c & d \\ -a & -b \end{array} \right] \quad
(c)\left[ \begin{array}{cc} a & b \\ a+c & b+d \end{array} \right] \quad
(d)\tfrac{1}{4}\left[ \begin{array}{cc} 4a & 4b \\ 3a+c & 3b+d \end{array} \right] \quad
(e)\left[ \begin{array}{cc} a & b \\ 4a & 4b \end{array} \right]
\end{equation*}

\vspace{6pt}

\item Find the determinant of the matrix: (a) $\left[ \begin{array}{rrrr} 2 & 3 & /4 & 5 \\ 0 & 1 & 3 & 5 \\ 2 & 3 & 4 & 6 \\ 4 & 5 & 6 & 7 \end{array} \right]$ \quad (b) $\left[ \begin{array}{rrrr} 1 & 3 & 0 & 0 \\ -2 & 1 & 0 & 0 \\ 0 & 0 & 4 & 6 \\ 0 & 0 & 6 & 7 \end{array} \right]$.

\vspace{6pt}

\item Let $\mathbf{u} = (2,5)$ and $\mathbf{v} = (-4,3)$ be vectors in $\R^2$.

\begin{enumerate}
\item Find the inner product $\mathbf{u} \cdot \mathbf{v}$. 
%\quad (b) Find the Euclidean distance $d(\mathbf{u},\mathbf{v})$ 
\quad (b) Are $\mathbf{u}$ and $\mathbf{v}$ orthogonal?
\end{enumerate}

\vspace{6pt}

\item Let $\R^4$ have the Euclidean inner product. Use the Gram-Schmidt process to transform the basis $\{(0,2,1,0),(1,-1,1,0),(1,2,1,0),(1,0,0,1)\}$ for $\R^4$ into an orthonormal basis.

\newpage

\item Let $T: \R^2 \to \R^2$ given by $w_1 = 3x_1 + 5x_2$, $w_2 = 2x_1 - 9x_2$.


\begin{enumerate}
\item Find the matrix for $T$ in the standard basis of $\R^2$. 
\item Calculate $T\left(\left[\begin{matrix}-2\\-3\end{matrix}\right]\right)$.
\item Is $T$ one-to-one? If so, find the matrix for the inverse linear transformation $T^{-1}$ in the standard basis.
\end{enumerate}


\vspace{6pt}

\item Find the matrix of $T$ in the standard bases matrix for the linear transformations $T$ defined by the following formulae:

\begin{enumerate}
\item $T(x_1, x_2, x_3, x_4) = (7x_1 + 2x_2 - x_3 + x_4, x_2 + x_3, -x_1)$.
\item $T(x_1, x_2, x_3, x_4) = (x_4, x_1, x_3, x_2, x_1 - x_3, x_2 + x_3)$.
\end{enumerate}

\vspace{6pt}

\item Let ${\cal M}_{22}$ be the vector space of $2 \times 2$ matrices with real number entries and with standard matrix addition and scalar multiplication. Determine which of the following are subspaces of ${\cal M}_{22}$.

\begin{enumerate}
\item $2 \times 2$ matrices $A$ with $\det A = 1$. 
\item $2 \times 2$ diagonal matrices.
\item $2 \times 2$ matrices with integer entries. 
\item $2 \times 2$ matrices $A$ such that $\mathrm{tr}(A) = 0$.
\item $2 \times 2$ matrices $A$ with nonzero entries.
\end{enumerate}

\vspace{6pt}

\item Determine which of the following are subspaces of $\R^3$.

\begin{enumerate}
\item Vectors of the form $(a, a, a)$. 
\item Vectors of the form $(a, 1, 1)$.
\item Vectors of the form $(a, b, c)$, where $b = a + c$.
\end{enumerate}

\vspace{6pt}

\item Bases and linear independence:
\begin{enumerate}
\item Is $\{(-3,0,4),(0,-1,2),(1,1,3)\}$ a linearly independent subset of $\R^3$?
\item Let $B = \{(3,9), (-4,a)\}$, find all values of $a$ so that $B$ becomes a basis for $\R^2$.
\item Let ${\cal B} = \{(2,-4),(3,8)\}$ be a basis for $\R^2$. Find the coordinate vector of $(1,1)$ relative to the basis ${\cal B}$.
\end{enumerate}

\vspace{6pt}

\item Determine whether the function (or mapping) is a linear transformation. Here, ${\cal M}_{mn}$ denotes the space of $m$ by $n$ matrices.
\begin{enumerate}
\item $T: {\cal M}_{nn} \to \R$, where $T(A) = \mathrm{tr}(A)$. 
\item $T: {\cal M}_{nn} \to {\cal M}_{nn}$, where $T(A) = -A$.
\item $T: {\cal M}_{mn} \to {\cal M}_{nm}$, where $T(A) = A^T$.
\item $T: Gl(n,\R) \to {\cal M}_{nn}$, where $T(A) = A^{-1}$, and $Gl(n,\R)$ is the set of $n\times n$ invertible matrices.
\end{enumerate}

\vspace{6pt}

\item Let $\mathbf{A} = \left[ \begin{array}{ll} 1 & 1 \\ 2 & 4 \end{array} \right]$.

\begin{enumerate}
\item Find all eigenvalues and corresponding eigenvectors of $\mathbf{A}$.
\item Find an invertible matrix $\mathbf{P}$ such that $\mathbf{P}^{-1}\mathbf{A}\mathbf{P}$ is a diagonal matrix.
\item Compute $\mathbf{A}^{30}$.
\end{enumerate}

\vspace{6pt}

\item Determine $T_1 \circ T_2 \circ T_3$ and evaluate $(T_1 \circ T_2 \circ T_3)(\sqrt{3}, -3)$ if $T_1: \R^2 \to \R^2$ is the counterclockwise rotation through an angle $\pi/4$, $T_2: \R^2 \to \R^2$ is the orthogonal projection on the $y$-axis and $T_3: \R^2 \to \R^2$ is the reflection about the $x$-axis.

\vspace{6pt}

\item Let the linear transformation $T: \R^3 \to \R^3$ be defined as $T(x_1, x_2, x_3) =$ $(7x_1 + 2x_2 - x_3, 2x_1 - 2x_2 + x_3, x_2 - x_3)$, and suppose ${\cal B} = \{(1,1,1),(0,1,1),(0,0,3)\}$ is a basis for $\R^3$. Find the matrix for $T$ with respect to the basis $B$. Is $T$ one-to-one? If so, find $T^{-1}$.

\vspace{6pt}


\item Let $A = \begin{bmatrix*}[r] 
2 & 2 & -1 & 0 & 1 \\ -1 & -1 & 2 & -3 & 0 \\ 1 & 1 & -2 & 0 & -1 \end{bmatrix*}$.

\begin{enumerate}
\item Find the bases for the rowspace, column space, and nullspace of $A$.
\item What are the rank($A$) and nullity($A$)?
\item If $T$ is the multiplication by the matrix $A$, what are bases for the range and the kernel of $T$?
\end{enumerate}

\vspace{6pt}

\item Fill in the blank.

\begin{enumerate}
\item A square matrix is not invertible iff at least one of its eigenvalues is \underline{\hspace{5cm}}.
\item A square matrix $A$ of order $n$ is diagonalizable iff $A$ has \underline{\phantom{xxx}} linearly independent eigenvectors.
\item A square matrix $A$ of order $n$ has $n$ distinct eigenvalues, then $A$ is \underline{\hspace{5cm}}.
\item A square matrix $A$ is diagonalizable iff the geometric multiplicity is equal to the \underline{\hspace{3cm}} for every eigenvalue.
\item Any bases for a vector space have the same \underline{\hspace{3.5cm}} of vectors.
\item If $A$ is a matrix with $n$ columns, then $\mathrm{rank}(A)\  +$ \underline{\hspace{2cm}} $= n$.
\item The basis for the column space of $A$ is determined by the columns containing \underline{\hspace{4cm}}.
\item If $\dim(V) = n$, and suppose a subset $B$ of $V$ has exactly $n$ vectors, then $B$ is a basis if either $B$ is linearly independent or \underline{\hspace{5cm}}.
\item[(e)] $W$ is a subspace of $V$ if \underline{\hspace{2cm}} $\in W$ for all vectors $u, v \in V$ and \underline{\hspace{2cm}} $\in W$ for all vectors $u \in W$ and scalars $k$.
\end{enumerate}

\vspace{6pt}

\item Let $T: \R^2 \to \R^3$ be defined by $T(x_1, x_2) = (x_1 + 2x_2, -x_1, 0)$. Find the matrix $[T]_{B',B}$ with respect to the bases $B = \{(1,3),(-2,4)\}$ and $B' = \{(1,1,1),(2,2,0),(3,0,0)\}$. Show $T(2,3) = (8,-2,3)$ by using the matrix multiplication.

\end{enumerate}
\vskip 2in
Updated 6/26 LF

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