For each archetype that is a system of equations, consider the corresponding homogeneous system of equations. Write elements of the solution set to these homogeneous systems in vector form, as guaranteed by Theorem VFSLS. Then write the null space of the coefficient matrix of each system as the span of a set of vectors, as described in Theorem SSNS.

Archetype A, Archetype B, Archetype C, Archetype D/Archetype E, Archetype F, Archetype G/Archetype H, Archetype I, Archetype J

Archetype K and Archetype L are defined as matrices. Use Theorem SSNS directly to find a set $S$ so that $\spn{S}$ is the null space of the matrix. Do not make any reference to the associated homogeneous system of equations in your solution.

Suppose that $S=\set{\colvector{2\\-1\\3\\4},\,\colvector{3\\2\\-2\\1}}$. Let $W=\spn{S}$ and let $\vect{x}=\colvector{5\\8\\-12\\-5}$. Is $\vect{x}\in W$? If so, provide an explicit linear combination that demonstrates this.

Suppose that $S=\set{\colvector{2\\-1\\3\\4},\,\colvector{3\\2\\-2\\1}}$. Let $W=\spn{S}$ and let $\vect{y}=\colvector{5\\1\\3\\5}$. Is $\vect{y}\in W$? If so, provide an explicit linear combination that demonstrates this.

Suppose $R=\set{\colvector{2\\-1\\3\\4\\0},\,\colvector{1\\1\\2\\2\\-1},\,\colvector{3\\-1\\0\\3\\-2}}$. Is $\vect{y}=\colvector{1\\-1\\-8\\-4\\-3}$ in $\spn{R}$?

Suppose $R=\set{\colvector{2\\-1\\3\\4\\0},\,\colvector{1\\1\\2\\2\\-1},\,\colvector{3\\-1\\0\\3\\-2}}$. Is $\vect{z}=\colvector{1\\1\\5\\3\\1}$ in $\spn{R}$?

Suppose that $S=\set{ \colvector{-1 \\ 2 \\ 1},\, \colvector{ 3 \\ 1 \\ 2},\, \colvector{ 1 \\ 5 \\ 4},\, \colvector{-6 \\ 5 \\ 1} }$. Let $W=\spn{S}$ and let $\vect{y}=\colvector{-5\\3\\0}$. Is $\vect{y}\in W$? If so, provide an explicit linear combination that demonstrates this.

Suppose that $S=\set{ \colvector{-1 \\ 2 \\ 1},\, \colvector{ 3 \\ 1 \\ 2},\, \colvector{ 1 \\ 5 \\ 4},\, \colvector{-6 \\ 5 \\ 1} }$. Let $W=\spn{S}$ and let $\vect{w}=\colvector{2\\1\\3}$. Is $\vect{w}\in W$? If so, provide an explicit linear combination that demonstrates this.

Let $A$ be the matrix below.

1. Find a set $S$ so that $\nsp{A}=\spn{S}$.
2. If $\vect{z}=\colvector{3 \\ -5 \\ 1 \\ 2}$, then show directly that $\vect{z}\in\nsp{A}$.
3. Write $\vect{z}$ as a linear combination of the vectors in $S$.

For the matrix $A$ below, find a set of vectors $S$ so that the span of $S$ equals the null space of $A$, $\spn{S}=\nsp{A}$. \begin{equation*} A= \begin{bmatrix} 1 & 1 & 6 & -8\\ 1 & -2 & 0 & 1\\ -2 & 1 & -6 & 7 \end{bmatrix} \end{equation*}

Consider the set of all size $2$ vectors in the Cartesian plane $\real{2}$.

1. Give a geometric description of the span of a single vector.
2. How can you tell if two vectors span the entire plane, without doing any row reduction or calculation?

Consider the set of all size $3$ vectors in Cartesian 3-space $\real{3}$.

1. Give a geometric description of the span of a single vector.
2. Describe the possibilities for the span of two vectors.
3. Describe the possibilities for the span of three vectors.

Let $\vect{u} = \colvector{1\\3\\-2}$ and $\vect{v} = \colvector{2\\-2\\1}$.

1. Find a vector $\vect{w}_1$, different from $\vect{u}$ and $\vect{v}$, so that $\spn{\set{\vect{u}, \vect{v}, \vect{w}_1}} = \spn{\set{\vect{u}, \vect{v}}}$.
2. Find a vector $\vect{w}_2$ so that $\spn{\set{\vect{u}, \vect{v}, \vect{w}_2}} \ne \spn{\set{\vect{u}, \vect{v}}}$.

In Example SCAD we began with the four columns of the coefficient matrix of Archetype D, and used these columns in a span construction. Then we methodically argued that we could remove the last column, then the third column, and create the same set by just doing a span construction with the first two columns. We claimed we could not go any further, and had removed as many vectors as possible. Provide a convincing argument for why a third vector cannot be removed.

In the spirit of Example SCAD, begin with the four columns of the coefficient matrix of Archetype C, and use these columns in a span construction to build the set $S$. Argue that $S$ can be expressed as the span of just three of the columns of the coefficient matrix (saying exactly which three) and in the spirit of Exercise SS.M20 argue that no one of these three vectors can be removed and still have a span construction create $S$.

Suppose that $\vect{v}_1,\,\vect{v}_2\in\complex{m}$. Prove that \begin{equation*} \spn{\set{\vect{v}_1,\,\vect{v}_2}}= \spn{\set{\vect{v}_1,\,\vect{v}_2,\,5\vect{v}_1+3\vect{v}_2}} \end{equation*}

Suppose that $S$ is a set of vectors from $\complex{m}$. Prove that the zero vector, $\zerovector$, is an element of $\spn{S}$.

Suppose that $S$ is a set of vectors from $\complex{m}$ and $\vect{x},\,\vect{y}\in\spn{S}$. Prove that $\vect{x}+\vect{y}\in\spn{S}$.

Suppose that $S$ is a set of vectors from $\complex{m}$, $\alpha\in\complex{\null}$, and $\vect{x}\in\spn{S}$. Prove that $\alpha\vect{x}\in\spn{S}$.