5 Vector Spaces Online Quiz Questions

Which description defines a basis for a vector space?

What is the dimension of the real polynomial space $P_2$, consisting of polynomials of degree at most 2?

Select all statements that are always true for sets of vectors in $\mathbb{R}^n$.

Let $v_1=(1,2,0)$ and $v_2=(0,1,1)$. Which vector belongs to $\operatorname{span}\{v_1,v_2\}$?

True or false: If the order of the vectors in an ordered basis changes, the coordinate vector of a fixed vector may also change.

Select all conclusions that necessarily follow when a set of vectors contains the zero vector.

What is $\dim(M_{2\times2})$, the vector space of all real $2\times2$ matrices?

Find a basis for the column space of $A=\begin{bmatrix}1&0&1\\0&1&1\\1&1&2\end{bmatrix}$, and explain why your chosen columns form a basis.

If $P_{\mathcal{B}}$ is the matrix whose columns are the vectors of an ordered basis $\mathcal{B}$, which equation reconstructs $v$ from its coordinate vector?

Which subset of $\mathbb{R}^3$ is not a subspace?

True or false: In an $n$-dimensional vector space, any set of exactly $n$ linearly independent vectors is a basis.

Which statement about the span of a set of vectors is necessarily true?

What is the dimension of the polynomial space $P_5$, consisting of real polynomials of degree at most 5?

Which conclusion follows for a set containing more than $n$ vectors in $\mathbb{R}^n$?

Let $v_1=(1,2,0)$, $v_2=(0,1,1)$, and $w=(3,8,2)$. If $w=av_1+bv_2$, what is the scalar coefficient $b$?

When row reduction identifies pivot columns for a matrix, which columns form a basis for the column space?

True or false: Let $W$ be a nonempty subset of a vector space $V$. If $cu+dv\in W$ for every $u,v\in W$ and all scalars $c,d$, then $W$ is a subspace of $V$.

Let $p(x)=1-2x+3x^2$ and $q(x)=4+x-x^2$. What is the coefficient of $x^2$ in $2p(x)-3q(x)$?