The archetypes listed below are matrices, or systems of equations with coefficient matrices. For each, compute the nullity and rank of the matrix. This information is listed for each archetype (along with the number of columns in the matrix, so as to illustrate Theorem RPNC), and notice how it could have been computed immediately after the determination of the sets $D$ and $F$ associated with the reduced row-echelon form of the matrix.
Archetype A, Archetype B, Archetype C, Archetype D/Archetype E, Archetype F, Archetype G/Archetype H, Archetype I, Archetype J, Archetype K, Archetype L

Find the dimension of the subspace $W = \setparts{\colvector{a + b\\ a + c\\a + d \\ d}}{a, b, c, d \in\complexes}$ of $\complex{4}$.

Find the dimension of the subspace $W = \setparts{a + bx + cx^2 + dx^3}{a + b + c + d = 0}$ of $P_3$.

Find the dimension of the subspace $W = \setparts{\begin{bmatrix} a & b\\c & d \end{bmatrix}}{a + b = c, b + c = d, c + d = a}$ of $M_{22}$.

For the matrix $A$ below, compute the dimension of the null space of $A$, $\dimension{\nsp{A}}$. \begin{equation*} A= \begin{bmatrix} 2 & -1 & -3 & 11 & 9 \\ 1 & 2 & 1 & -7 & -3 \\ 3 & 1 & -3 & 6 & 8 \\ 2 & 1 & 2 & -5 & -3 \end{bmatrix} \end{equation*}

The set $W$ below is a subspace of $\complex{4}$. Find the dimension of $W$. \begin{equation*} W=\spn{\set{ \colvector{2\\-3\\4\\1},\, \colvector{3\\0\\1\\-2},\, \colvector{-4\\-3\\2\\5} }} \end{equation*}

Find the rank and nullity of the matrix $A = \begin{bmatrix} 1 & 0 & 1\\ 1 & 2 & 2\\ 2 & 1 & 1\\ -1 & 0 & 1\\ 1 & 1 & 2 \end{bmatrix}$.

Find the rank and nullity of the matrix $A = \begin{bmatrix} 1 & 2 & 1 & 1 & 1\\ 1 & 3 & 2 & 0 & 4\\ 1 & 2 & 1 & 1 & 1 \end{bmatrix}$.

Find the rank and nullity of the matrix $A = \begin{bmatrix} 3 & 2 & 1 & 1 & 1\\ 2 & 3 & 0 & 1 & 1\\ -1 & 1 & 2 & 1 & 0\\ 1 & 1 & 0 & 1 & 1\\ 0 & 1 & 1 & 2 & -1 \end{bmatrix}$.

In Example LDP4 we determined that the set of five polynomials, $T$, is linearly dependent by a simple invocation of Theorem SSLD. Prove that $T$ is linearly dependent from scratch, beginning with Definition LI.

$M_{22}$ is the vector space of $2\times 2$ matrices. Let $S_{22}$ denote the set of all $2\times 2$ symmetric matrices. That is \begin{equation*} S_{22}=\setparts{A\in M_{22}}{\transpose{A}=A} \end{equation*}

1. Show that $S_{22}$ is a subspace of $M_{22}$.
2. Exhibit a basis for $S_{22}$ and prove that it has the required properties.
3. What is the dimension of $S_{22}$?

A $2\times 2$ matrix $B$ is upper triangular if $\matrixentry{B}{21}=0$. Let $UT_2$ be the set of all $2\times 2$ upper triangular matrices. Then $UT_2$ is a subspace of the vector space of all $2\times 2$ matrices, $M_{22}$ (you may assume this). Determine the dimension of $UT_2$ providing all of the necessary justifications for your answer.