Linear Algebra Self Test

Linear Algebra Quiz

Test your knowledge of vector spaces, transformations, and matrices

Q1

A vector space $V$ over a field $\mathbb{F}$ is defined by:

A set with vector addition and scalar multiplication satisfying 8 axioms (closure, associativity, commutativity, identity, inverse, and distributive laws) A set with vector addition and scalar multiplication satisfying closure and associativity A set with vector addition and scalar multiplication satisfying commutativity and distributive laws A set with vector addition and scalar multiplication satisfying identity and inverse properties

Q2

A set of vectors $\{v_1, v_2, \ldots, v_k\}$ in a vector space $V$ is linearly dependent if:

There exist scalars $c_1, c_2, \ldots, c_k$ not all zero such that $c_1v_1 + c_2v_2 + \cdots + c_kv_k = 0$ There exist scalars $c_1, c_2, \ldots, c_k$ such that $c_1v_1 + c_2v_2 + \cdots + c_kv_k = 0$ Each vector can be written as a linear combination of the others Both (a) and (c)

Q3

Let $V = \mathbb{R}^3$ and consider the vectors $v_1 = (1, 2, 3)$, $v_2 = (4, 5, 6)$, $v_3 = (7, 8, 9)$. Which statement is true?

The vectors are linearly dependent and span a 2-dimensional subspace The vectors are linearly independent and span $\mathbb{R}^3$ The vectors are linearly dependent but span $\mathbb{R}^3$ The vectors are linearly independent but span a 2-dimensional subspace

Q4

A subset $W$ of a vector space $V$ is a subspace if:

$W$ is non-empty and closed under vector addition and scalar multiplication $W$ contains the zero vector and is closed under vector addition $W$ is closed under vector addition and scalar multiplication $W$ is non-empty and closed under linear combinations

Q5

A basis for a vector space $V$ is:

A linearly independent set that spans $V$ A spanning set for $V$ A linearly independent set in $V$ A maximal linearly independent set in $V$

Q6

The dimension of a vector space $V$ is:

The number of vectors in any basis for $V$ The maximum number of linearly independent vectors in $V$ The minimum number of vectors needed to span $V$ Both (a) and (b)

Q7

Let $V$ be the vector space of all polynomials with real coefficients. What is $\dim(V)$?

$\infty$ Finite but unbounded 0$ Not defined

Q8

Which of the following statements is FALSE?

Every vector space has a basis All bases for a vector space have the same number of elements A linearly independent set can always be extended to a basis Every spanning set contains a basis

Q9

Let $V = \mathbb{R}^2$ and $W_1 = \{(x, 0) : x \in \mathbb{R}\}$, $W_2 = \{(0, y) : y \in \mathbb{R}\}$. What is $\dim(W_1 + W_2)$?

$2$ $1$ $0$ $\infty$

Q10

Which of the following is NOT a vector space over $\mathbb{C}$?

$\mathbb{C}^n$ with component-wise addition and scalar multiplication The set of all $n \times n$ Hermitian matrices The set of all $n \times n$ unitary matrices The set of all $n \times n$ invertible matrices

Q11

A function $T: V \to W$ between vector spaces is a linear transformation if:

$T(u + v) = T(u) + T(v)$ and $T(cu) = cT(u)$ for all $u, v \in V$ and $c \in \mathbb{F}$ $T(u + v) = T(u) + T(v)$ for all $u, v \in V$ $T(cu) = cT(u)$ for all $u \in V$ and $c \in \mathbb{F}$ $T$ is bijective

Q12

For a linear transformation $T: V \to W$, the rank-nullity theorem states that:

$\text{rank}(T) + \text{nullity}(T) = \dim(V)$ $\text{rank}(T) + \text{nullity}(T) = \dim(W)$ $\text{rank}(T) = \text{nullity}(T)$ $\text{rank}(T) \cdot \text{nullity}(T) = \dim(V)$

Q13

If $T: V \to W$ is a linear transformation and $B_V = \{v_1, \ldots, v_n\}$, $B_W = \{w_1, \ldots, w_m\}$ are bases for $V$ and $W$ respectively, then the matrix representation $[T]_{B_V}^{B_W}$ has:

Columns that are the coordinate vectors of $T(v_j)$ with respect to $B_W$ Rows that are the coordinate vectors of $T(v_j)$ with respect to $B_W$ Columns that are the coordinate vectors of $T(w_i)$ with respect to $B_V$ Rows that are the coordinate vectors of $T(w_i)$ with respect to $B_V$

Q14

A linear functional on a vector space $V$ is:

A linear transformation from $V$ to its field of scalars A linear transformation from the field of scalars to $V$ A linear transformation from $V$ to itself A basis for the dual space $V^*$

Q15

Let $T: \mathbb{R}^3 \to \mathbb{R}^2$ be defined by $T(x, y, z) = (x + y, y + z)$. What is $\text{nullity}(T)$?

$1$ $2$ $0$ $3$

Q16

The row echelon form of a matrix is characterized by:

All nonzero rows are above rows of zeros, and the leading coefficient of a nonzero row is always strictly to the right of the leading coefficient of the row above it All nonzero rows are below rows of zeros, and the leading coefficient of a nonzero row is always strictly to the right of the leading coefficient of the row above it The matrix is upper triangular with 1's on the diagonal The matrix is diagonal with nonzero entries on the diagonal

Q17

An $n \times n$ matrix $A$ is invertible if and only if:

$\det(A) \neq 0$ $A$ has full rank The homogeneous system $Ax = 0$ has only the trivial solution All of the above

Q18

Let $A$ be an $m \times n$ matrix with rank $r$. Which statement is FALSE?

$r \leq \min(m, n)$ The row rank equals the column rank $r = \max(m, n)$ if $A$ is full rank $r$ is the dimension of the column space of $A$

Q19

Two $n \times n$ matrices $A$ and $B$ are similar if:

There exists an invertible matrix $P$ such that $B = P^{-1}AP$ There exists an invertible matrix $P$ such that $B = PAP^{-1}$ $A$ and $B$ have the same eigenvalues $A$ and $B$ have the same determinant

Q20

The system $Ax = b$ has a solution if and only if:

$\text{rank}(A) = \text{rank}([A|b])$ where $[A|b]$ is the augmented matrix $\text{rank}(A) = \text{rank}(A)$ $\det(A) \neq 0$ $b$ is in the column space of $A$

Q21

An eigenvalue of an $n \times n$ matrix $A$ is:

A scalar $\lambda$ such that there exists a nonzero vector $v$ with $Av = \lambda v$ A scalar $\lambda$ such that $\det(A - \lambda I) = 0$ A root of the characteristic polynomial of $A$ All of the above

Q22

The characteristic polynomial of an $n \times n$ matrix $A$ is:

$p(\lambda) = \det(A - \lambda I)$ $p(\lambda) = \det(\lambda I - A)$ $p(\lambda) = \det(A + \lambda I)$ $p(\lambda) = \det(\lambda A - I)$

Q23

The Cayley-Hamilton theorem states that:

Every square matrix satisfies its own characteristic equation Every square matrix has eigenvalues Every square matrix is diagonalizable Every square matrix has a full set of eigenvectors

Q24

Let $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$. Which statement is true?

$A$ has eigenvalue 0 with algebraic multiplicity 2 and geometric multiplicity 1 $A$ has eigenvalue 0 with algebraic multiplicity 1 and geometric multiplicity 1 $A$ has eigenvalues 0 and 1 $A$ has no eigenvalues

Q25

An $n \times n$ matrix $A$ is diagonalizable if:

$A$ has $n$ linearly independent eigenvectors $A$ has $n$ distinct eigenvalues The geometric multiplicity equals the algebraic multiplicity for each eigenvalue Both (a) and (c)

Q26

Which of the following is true for a real symmetric matrix?

All eigenvalues are real and it is diagonalizable All eigenvalues are real but it may not be diagonalizable It is diagonalizable but eigenvalues may be complex It has no real eigenvalues

Q27

A real matrix $Q$ is orthogonal if:

$Q^T Q = I$ $Q Q^T = I$ $Q^{-1} = Q^T$ All of the above

Q28

A complex matrix $U$ is unitary if:

$U^* U = I$ where $U^*$ is the conjugate transpose $U U^* = I$ where $U^*$ is the conjugate transpose $U^{-1} = U^T$ Both (a) and (b)

Q29

A quadratic form in $n$ variables is:

A homogeneous polynomial of degree 2 in $n$ variables A polynomial of degree 2 in $n$ variables A homogeneous polynomial of degree at most 2 in $n$ variables A polynomial that can be written as $x^T A x$ for some matrix $A$

Q30

Which of the following matrices is NOT diagonalizable over $\mathbb{R}$?

$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ $\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}$ $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$