Practice Test 1 - Linear Algebra

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. Let $V$ be the set of positive real numbers, $\mathbb{R}^+$. Define 'vector addition' as standard multiplication ($x \oplus y = xy$) and 'scalar multiplication' as standard multiplication ($c \odot x = cx$) for $c \in \mathbb{R}$. Why is $V$ not a vector space with these operations?
2. Let $T: \mathbb{R}^2 \to \mathbb{R}^3$ be a linear transformation such that $T\left(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\right) = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$ and $T\left(\begin{bmatrix} 0 \\ 1 \end{bmatrix}\right) = \begin{bmatrix} 0 \\ 4 \\ 1 \end{bmatrix}$. What is the value of $T\left(\begin{bmatrix} 3 \\ -2 \end{bmatrix}\right)$?
3. Let $W$ be a subspace of $\mathbb{R}^3$ and let $\vec{y}$ be a vector in $\mathbb{R}^3$. If the orthogonal projection of $\vec{y}$ onto $W$ is the zero vector (i.e., $\mathrm{proj}_W(\vec{y}) = \vec{0}$), what must be true about $\vec{y}$?
4. Let $D(\vec{r}_1, \vec{r}_2, \vec{r}_3)$ denote the determinant of a $3 \times 3$ matrix with rows $\vec{r}_1, \vec{r}_2, \vec{r}_3$. Which of the following expressions is a correct statement of the linearity property of determinants?
5. The normal equations $A^T A \hat{\mathbf{x}} = A^T \mathbf{b}$ produce the set of all least-squares solutions. Under what geometric condition will there be infinitely many least-squares solutions $\hat{\mathbf{x}}$?
6. Given the following linear transformations: $T_1: \mathbb{R}^3 \to \mathbb{R}^2$ $T_2: \mathbb{R}^2 \to \mathbb{R}^3$ $T_3: \mathbb{R}^3 \to \mathbb{R}^3$ $T_4: \mathbb{R}^2 \to \mathbb{R}^2$ Which of the following composite transformations is NOT well-defined?
7. Consider the subspace $V$ of $\mathbb{R}^6$ spanned by the vectors $u_1 = (1, 0, 1, 1, 0, 1)$, $u_2 = (0, 1, 1, 0, 1, 1)$, $u_3 = (1, 1, 2, 1, 1, 2)$, $u_4 = (2, 1, 3, 2, 1, 3)$, and $u_5 = (1, 2, 3, 1, 2, 3)$. After determining linear dependencies, which subset forms a basis for $V$?
8. Consider the matrix $A = \begin{pmatrix} 0 & 3 \\ 1 & 5 \\ 0 & 4 \end{pmatrix}$. Let $A=QR$ be its QR factorization. What are the first column of $Q$, denoted $\mathbf{q}_1$, and the entry $r_{11}$ of $R$?
9. Let $A$ be a $4 \times 4$ matrix whose columns are $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$. If it is known that $3\mathbf{v}_1 - \mathbf{v}_3 = \mathbf{0}$, which of the following statements must be true?
10. Consider the orthonormal basis $\{\mathbf{q}_1, \mathbf{q}_2, \mathbf{q}_3\}$ for $\mathbb{R}^3$ where $\mathbf{q}_1 = \frac{1}{3}(2, 1, 2)$, $\mathbf{q}_2 = \frac{1}{3}(1, 2, -2)$, and $\mathbf{q}_3 = \frac{1}{3}(2, -2, 1)$. If vector $\mathbf{v}$ satisfies $\|\mathbf{v}\|^2 = 14$ and has coordinates $(\alpha, \beta, \gamma)$ in this basis, what is $\alpha^2 + \beta^2 + \gamma^2$?
11. Let $A$ be a $3 \times 3$ matrix with eigenvalues $\lambda=0, 2, 3$. Which of the following matrices is guaranteed to be similar to $A$?
12. Consider the matrix $A = \begin{pmatrix} 3 & a & b \\ 0 & 3 & c \\ 0 & 0 & 5 \end{pmatrix}$ where $a, b, c$ are nonzero real numbers. If $\det(A - 3I) = 0$, what can be concluded about the geometric multiplicity of eigenvalue $3$?
13. A linear transformation $T: \mathbb{R}^4 \to \mathbb{R}^3$ is represented by a $3 \times 4$ matrix $A$. If the transformation $T$ is surjective (onto), what must be true about the reduced row echelon form (RREF) of $A$?
14. For which value of the scalar $k$ does the set of vectors $S = \{\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ -2 \\ k \end{pmatrix}\}$ not form a generating set for $\mathbb{R}^3$?
15. A system's state is described by the vector $\vec{x}_k$ at time $k$, and it evolves according to $\vec{x}_{k+1} = P\vec{x}_k$ with $P = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix}$. If the initial state is $\vec{x}_0 = \begin{pmatrix} 0.8 \\ 0.2 \end{pmatrix}$, what is the state vector $\vec{x}_2$?
16. A direct justification for why any $n \times n$ column-stochastic matrix $P$ must have an eigenvalue of $\lambda = 1$ is that:
17. A linear system's solution is given in parametric form as $\vec{x} = \begin{pmatrix} 4 \\ 0 \\ -1 \\ 0 \end{pmatrix} + s \begin{pmatrix} -2 \\ 1 \\ 0 \\ 0 \end{pmatrix} + t \begin{pmatrix} 0 \\ 0 \\ 3 \\ 1 \end{pmatrix}$. Which of the following could be the reduced row echelon form of the system's augmented matrix?
18. Let $T: \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation. If $n > m$ and $T$ is known to be surjective, what must be the dimension of the kernel of $T$?
19. A $3 \times 3$ real matrix has eigenvalues $\lambda_1 = 2$, $\lambda_2 = -1 + 3i$, and $\lambda_3 = -1 - 3i$. If the matrix represents a dynamical system $\mathbf{x}' = A\mathbf{x}$, which statement best describes the long-term behavior of solutions?
20. Let $\mathcal{B} = \{\vec{b}_1, \vec{b}_2\}$ and $\mathcal{C} = \{\vec{c}_1, \vec{c}_2\}$ be two bases for a vector space $V$. Let $P$ be the change-of-coordinates matrix from $\mathcal{B}$ to $\mathcal{C}$, denoted $P_{\mathcal{C} \leftarrow \mathcal{B}}$. Which of the following correctly describes the columns of $P$?

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