Math Exercises

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\fancyhead[L]{\textbf{MATH 340 --- Linear Algebra}}
\fancyhead[R]{\textit{Practice Problems}}
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\begin{document}

\begin{center}
  {\LARGE\bfseries MATH 340 --- Linear Algebra}\\[6pt]
  {\Large Practice Exercises}\\[4pt]
  {\large Chapters 1--4 Review}\\[8pt]
  \textcolor{gray}{\small Difficulty: \difficulty{E} Easy \quad \difficulty{M} Medium \quad \difficulty{H} Hard}
\end{center}

\vspace{8pt}
\hrule
\vspace{12pt}

% ============================================================
\section{Vector Spaces and Subspaces}

\exercise[E]
Determine whether $W = \{(x,y,z) \in \mathbb{R}^3 : x + 2y - z = 0\}$ is a subspace of $\mathbb{R}^3$. Prove your answer.

\exercise[E]
Let $\mathbf{u} = (1, -2, 3)$ and $\mathbf{v} = (4, 0, -1)$. Compute $2\mathbf{u} - 3\mathbf{v}$ and $\|\mathbf{u} + \mathbf{v}\|$.

\exercise[M]
Prove that the set $W = \{p(x) \in P_3 : p(1) = 0\}$ is a subspace of $P_3$ (polynomials of degree $\le 3$). Find a basis for $W$ and determine $\dim(W)$.

\exercise[M]
Let $V = M_{2\times 2}(\mathbb{R})$ (the space of $2 \times 2$ real matrices). Show that the set of symmetric matrices $S = \{A \in V : A^T = A\}$ is a subspace. What is $\dim(S)$?

\exercise[H]
Prove that $\mathbb{R}^n$ cannot be written as the union of finitely many proper subspaces.

\vspace{6pt}

% ============================================================
\section{Linear Independence and Bases}

\exercise[E]
Determine whether the vectors $\{(1,0,2),\, (0,1,1),\, (1,1,3)\}$ are linearly independent in $\mathbb{R}^3$.

\exercise[E]
Express $(7, 4, -1)$ as a linear combination of $\{(1,0,1),\, (0,1,2),\, (2,1,0)\}$, or show that it is not possible.

\exercise[M]
Find a basis for the column space and the null space of
\[
A = \begin{pmatrix} 1 & 2 & -1 & 3 \\ 2 & 4 & 1 & 0 \\ 0 & 0 & 3 & -6 \end{pmatrix}.
\]
Verify the rank-nullity theorem for this matrix.

\exercise[M]
Let $V = \text{span}\{1, \cos^2 x, \sin^2 x, \cos 2x\}$ as a subspace of the vector space of real-valued functions on $\mathbb{R}$. Find $\dim(V)$ and exhibit a basis.

\exercise[H]
Let $V$ be a finite-dimensional vector space and let $W_1, W_2$ be subspaces. Prove the dimension formula:
\[
\dim(W_1 + W_2) = \dim(W_1) + \dim(W_2) - \dim(W_1 \cap W_2).
\]

\vspace{6pt}

% ============================================================
\section{Linear Transformations}

\exercise[E]
Let $T : \mathbb{R}^2 \to \mathbb{R}^2$ be defined by $T(x,y) = (2x+y,\, x-3y)$. Find the standard matrix of $T$ and compute $T(3, -1)$.

\exercise[M]
Let $T : \mathbb{R}^3 \to \mathbb{R}^2$ be given by $T(x,y,z) = (x+y,\, y+z)$. Find $\ker(T)$ and $\text{Im}(T)$. Verify the rank-nullity theorem.

\exercise[M]
Show that the map $T : P_2 \to P_2$ defined by $T(p)(x) = p'(x) + xp(0)$ is a linear transformation. Find its matrix representation with respect to the basis $\{1, x, x^2\}$.

\exercise[H]
Let $V$ be a finite-dimensional vector space and $T : V \to V$ a linear operator such that $T^2 = T$ (idempotent). Prove that $V = \ker(T) \oplus \text{Im}(T)$.

\vspace{6pt}

% ============================================================
\section{Eigenvalues and Diagonalization}

\exercise[E]
Find the eigenvalues and eigenvectors of $A = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}$.

\exercise[M]
Determine whether the matrix
\[
B = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{pmatrix}
\]
is diagonalizable. Justify your answer.

\exercise[M]
Let $A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Compute $A^{100}$. \textit{Hint: Consider the eigenvalues of $A$.}

\exercise[H]
Let $A$ and $B$ be $n \times n$ matrices. Prove that $AB$ and $BA$ have the same eigenvalues. \textit{Hint: If $\lambda \neq 0$ is an eigenvalue of $AB$ with eigenvector $\mathbf{v}$, consider $B\mathbf{v}$.}

\exercise[H]
Let $A$ be a real symmetric $n \times n$ matrix. Prove that eigenvectors corresponding to distinct eigenvalues are orthogonal.

\vspace{20pt}

% ============================================================
\begin{hintbox}
\begin{description}[font=\bfseries,leftmargin=!,labelwidth=3cm,itemsep=3pt]
  \item[Exercise 1.1] Check the three subspace criteria: zero vector, closure under addition, closure under scalar multiplication.
  \item[Exercise 1.3] Consider polynomials of the form $(x-1)q(x)$ where $\deg q \le 2$.
  \item[Exercise 1.5] Consider a vector not in any of the subspaces and use scalar multiples.
  \item[Exercise 2.3] Row reduce the augmented matrix $[A \mid 0]$ and use pivot/free variable analysis.
  \item[Exercise 2.4] Note the identity $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1$.
  \item[Exercise 2.5] Define a linear map $\phi: W_1 \times W_2 \to W_1+W_2$ and use the first isomorphism theorem.
  \item[Exercise 3.4] Show $\ker(T) \cap \text{Im}(T) = \{0\}$ and use dimension counting.
  \item[Exercise 4.3] Find a pattern in $A^2, A^3, A^4, \ldots$ or diagonalize over $\mathbb{C}$.
  \item[Exercise 4.4] For the case $\lambda = 0$, treat it separately.
  \item[Exercise 4.5] Use the fact that $A = A^T$ and consider $\lambda_1 \langle \mathbf{v}_1, \mathbf{v}_2 \rangle$.
\end{description}
\end{hintbox}

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