Working within the vector space $\complex{3}$, determine if $\vect{b} = \colvector{4\\3\\1}$ is in the subspace $W$, \begin{equation*} W = \spn{\set{ \colvector{3\\2\\3}, \colvector{1\\0\\3}, \colvector{1\\1\\0}, \colvector{2\\1\\3} }} \end{equation*}

Working within the vector space $\complex{4}$, determine if $\vect{b} = \colvector{1\\1\\0\\1}$ is in the subspace $W$, \begin{equation*} W =\spn{\set{ \colvector{1\\2\\-1\\1}, \colvector{1\\0\\3\\1}, \colvector{2\\1\\1\\2} }} \end{equation*}

Working within the vector space $\complex{4}$, determine if $\vect{b} = \colvector{2\\1\\2\\1}$ is in the subspace $W$, \begin{equation*} W = \spn{\set{ \colvector{1\\2\\0\\2}, \colvector{1\\0\\3\\1}, \colvector{0\\1\\0\\2}, \colvector{1\\1\\2\\0} }} \end{equation*}

Working within the vector space $P_3$ of polynomials of degree 3 or less, determine if $p(x)=x^3+6x+4$ is in the subspace $W$ below. \begin{equation*} W=\spn{\set{x^3+x^2+x,\,x^3+2x-6,\,x^2-5}} \end{equation*}

Consider the subspace \begin{equation*} W=\spn{\set{ \begin{bmatrix} 2 & 1\\3 & -1 \end{bmatrix} ,\, \begin{bmatrix} 4 & 0\\2 & 3 \end{bmatrix} ,\, \begin{bmatrix} -3 & 1\\2 & 1 \end{bmatrix} }} \end{equation*} of the vector space of $2\times 2$ matrices, $M_{22}$. Is $ C= \begin{bmatrix} -3 & 3\\6 & -4 \end{bmatrix} $ an element of $W$?

Show that the set $W=\setparts{\colvector{x_1\\x_2}}{3x_1-5x_2=12}$ from Example NSC2Z fails Property AC and Property SC.

Show that the set $Y=\setparts{\colvector{x_1\\x_2}}{x_1\in{\mathbb Z},\,x_2\in{\mathbb Z}}$ from Example NSC2S has Property AC.

In $\complex{3}$, the vector space of column vectors of size 3, prove that the set $Z$ is a subspace. \begin{equation*} Z=\setparts{\colvector{x_1\\x_2\\x_3}}{4x_1-x_2+5x_3=0} \end{equation*}

A square matrix $A$ of size $n$ is upper triangular if $\matrixentry{A}{ij}=0$ whenever $i>j$. Let $UT_n$ be the set of all upper triangular matrices of size $n$. Prove that $UT_n$ is a subspace of the vector space of all square matrices of size $n$, $M_{nn}$.

Let $P$ be the set of all polynomials, of any degree. The set $P$ is a vector space. Let $E$ be the subset of $P$ consisting of all polynomials with only terms of even degree. Prove or disprove: the set $E$ is a subspace of $P$.

Let $P$ be the set of all polynomials, of any degree. The set $P$ is a vector space. Let $F$ be the subset of $P$ consisting of all polynomials with only terms of odd degree. Prove or disprove: the set $F$ is a subspace of $P$.