Let $T$ be the set of columns of the matrix $B$ below. Define $W=\spn{T}$. Find a set $R$ so that (1) $R$ has 3 vectors, (2) $R$ is a subset of $T$, and (3) $W=\spn{R}$. \begin{equation*} B= \begin{bmatrix} -3 & 1 & -2 & 7\\ -1 & 2 & 1 & 4\\ 1 & 1 & 2 & -1 \end{bmatrix} \end{equation*}

Verify that the set $R^\prime=\set{\vect{v}_1,\,\vect{v}_2,\,\vect{v}_4}$ at the end of Example RSC5 is linearly independent.

Consider the set of vectors from $\complex{3}$, $W$, given below. Find a linearly independent set $T$ that contains three vectors from $W$ and such that $\spn{W}=\spn{T}$. \begin{equation*} W= \set{\vect{v}_1,\,\vect{v}_2,\,\vect{v}_3,\,\vect{v}_4,\,\vect{v}_5} =\set{ \colvector{2\\1\\1},\, \colvector{-1\\-1\\1},\, \colvector{1\\2\\3},\, \colvector{3\\1\\3},\, \colvector{0\\1\\-3} } \end{equation*}

Given the set $S$ below, find a linearly independent set $T$ so that $\spn{T}=\spn{S}$. \begin{equation*} S=\set{ \colvector{2\\-1\\2},\, \colvector{3\\0\\1},\, \colvector{1\\1\\-1},\, \colvector{5\\-1\\3} } \end{equation*}

Let $W$ be the span of the set of vectors $S$ below, $W=\spn{S}$. Find a set $T$ so that 1) the span of $T$ is $W$, $\spn{T}=W$, (2) $T$ is a linearly independent set, and (3) $T$ is a subset of $S$. \begin{align*} S&= \set{ \colvector{1 \\ 2 \\ -1},\, \colvector{2 \\ -3 \\ 1},\, \colvector{4 \\ 1 \\ -1},\, \colvector{3 \\ 1 \\ 1},\, \colvector{3 \\ -1 \\ 0} } \end{align*}

Let $T$ be the set of vectors $T=\set{ \colvector{1 \\ -1 \\ 2},\, \colvector{3 \\ 0 \\ 1},\, \colvector{4 \\ 2 \\ 3},\, \colvector{3 \\ 0 \\ 6} }$. Find two different subsets of $T$, named $R$ and $S$, so that $R$ and $S$ each contain three vectors, and so that $\spn{R}=\spn{T}$ and $\spn{S}=\spn{T}$. Prove that both $R$ and $S$ are linearly independent.

Reprise Example RES by creating a new version of the vector $\vect{y}$. In other words, form a new, different linear combination of the vectors in $R$ to create a new vector $\vect{y}$ (but do not simplify the problem too much by choosing any of the five new scalars to be zero). Then express this new $\vect{y}$ as a combination of the vectors in $P$.

At the conclusion of Example RSC4 two alternative solutions, sets $T^{\prime}$ and $T^{*}$, are proposed. Verify these claims by proving that $\spn{T}=\spn{T^{\prime}}$ and $\spn{T}=\spn{T^{*}}$.

Suppose that $\vect{v}_1$ and $\vect{v}_2$ are any two vectors from $\complex{m}$. Prove the following set equality. \begin{equation*} \spn{\set{\vect{v}_1,\,\vect{v}_2}} = \spn{\set{\vect{v}_1+\vect{v}_2,\,\vect{v}_1-\vect{v}_2}} \end{equation*}