2. Linear Algebra

Vectors And Spaces — Quiz

Test your understanding of vectors and spaces with 5 practice questions.

Practice Questions

Question 1

Given a set of vectors $\mathbf{S} = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ in a vector space $\mathbf{V}$, what is the most precise condition for $\mathbf{S}$ to be a basis for $\mathbf{V}$?

Question 2

Consider the vector space $\mathbf{P}_2$ of all polynomials of degree at most 2. Which of the following sets is a basis for $\mathbf{P}_2$?

Question 3

Let $\mathbf{W}$ be the set of all vectors in $\mathbb{R}^3$ of the form $\begin{pmatrix} a \ b \ a+b \end{pmatrix}$. Is $\mathbf{W}$ a subspace of $\mathbb{R}^3$?

Question 4

What is the dimension of the vector space of all $3 \times 2$ matrices, denoted as $M_{3,2}(\mathbb{R})$?

Question 5

Given the vectors $\mathbf{v}_1 = \begin{pmatrix} 1 \ 2 \end{pmatrix}$ and $\mathbf{v}_2 = \begin{pmatrix} 3 \ 6 \end{pmatrix}$ in $\mathbb{R}^2$, are they linearly independent or linearly dependent?