Vector Spaces and Linear Systems

Transpose each.

• (a)  $\begin{pmatrix}2&1\\ 3&1\end{pmatrix}$     (b)  $\begin{pmatrix}2&1\\ 1&3\end{pmatrix}$     (c)  $\begin{pmatrix}1&4&3\\ 6&7&8\end{pmatrix}$     (d)  $\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}$     (e)  $(\begin{matrix}-1&-2\end{matrix})$

Decide if the vector is in the row space of the matrix.

• (a)  $\begin{pmatrix}2&1\\ 3&1\end{pmatrix}$, $(\begin{matrix}1&0\end{matrix})$     (b)  $\begin{pmatrix}0&1&3\\ -1&0&1\\ -1&2&7\end{pmatrix}$, $(\begin{matrix}1&1&1\end{matrix})$

Decide if the vector is in the column space.

• (a)  $\begin{pmatrix}1&1\\ 1&1\end{pmatrix}$, $\begin{pmatrix}1\\ 3\end{pmatrix}$     (b)  $\begin{pmatrix}1&3&1\\ 2&0&4\\ 1&-3&-3\end{pmatrix}$, $\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}$

Decide if the vector is in the column space of the matrix.

• (a)  $\begin{pmatrix}2&1\\ 2&5\end{pmatrix}$, $\begin{pmatrix}1\\ -3\end{pmatrix}$     (b)  $\begin{pmatrix}4&-8\\ 2&-4\end{pmatrix}$, $\begin{pmatrix}0\\ 1\end{pmatrix}$     (c)  $\begin{pmatrix}1&-1&1\\ 1&1&-1\\ -1&-1&1\end{pmatrix}$, $\begin{pmatrix}2\\ 0\\ 0\end{pmatrix}$

Find a basis for the row space of this matrix.

Find the rank of each matrix.

• (a)  $\begin{pmatrix}2&1&3\\ 1&-1&2\\ 1&0&3\end{pmatrix}$     (b)  $\begin{pmatrix}1&-1&2\\ 3&-3&6\\ -2&2&-4\end{pmatrix}$     (c)  $\begin{pmatrix}1&3&2\\ 5&1&1\\ 6&4&3\end{pmatrix}$     (d)  $\begin{pmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$

Give a basis for the column space of this matrix. Give the matrix’s rank.

Find a basis for the span of each set.

• $\text{$\{\hskip 1.00006pt(\begin{matrix}1&3\end{matrix}),(\begin{matrix}-1&3\end{matrix}),(\begin{matrix}1&4\end{matrix}),(\begin{matrix}2&1\end{matrix})\hskip 1.00006pt\}$}\subseteq\mathcal{M}_{1\!\times\!2}$

• $\text{$\{\hskip 1.00006pt\begin{pmatrix}1\\ 2\\ 1\end{pmatrix},\begin{pmatrix}3\\ 1\\ -1\end{pmatrix},\begin{pmatrix}1\\ -3\\ -3\end{pmatrix}\hskip 1.00006pt\}$}\subseteq\mathbb{R}^{3}$

• $\text{$\{\hskip 1.00006pt1+x,1-x^{2},3+2x-x^{2}\hskip 1.00006pt\}$}\subseteq\mathcal{P}_{3}$

• $\text{$\{\hskip 1.00006pt\begin{pmatrix}1&0&1\\ 3&1&-1\end{pmatrix},\begin{pmatrix}1&0&3\\ 2&1&4\end{pmatrix},\begin{pmatrix}-1&0&-5\\ -1&-1&-9\end{pmatrix}\hskip 1.00006pt\}$}\subseteq\mathcal{M}_{2\!\times\!3}$

Give a basis for the span of each set, in the natural vector space.

• $\{\hskip 1.00006pt\begin{pmatrix}1\\ 1\\ 3\end{pmatrix},\begin{pmatrix}-1\\ 2\\ 0\end{pmatrix},\begin{pmatrix}0\\ 12\\ 6\end{pmatrix}\hskip 1.00006pt\}$

• $\{\hskip 1.00006ptx+x^{2},2-2x,7,4+3x+2x^{2}\hskip 1.00006pt\}$

Which matrices have rank zero? Rank one?

Given $a,b,c\in\mathbb{R}$, what choice of $d$ will cause this matrix to have the rank of one?

Find the column rank of this matrix.

Show that a linear system with at least one solution has at most one solution if and only if the matrix of coefficients has rank equal to the number of its columns.

If a matrix is $5\!\times\!9$, which set must be dependent, its set of rows or its set of columns?

Give an example to show that, despite that they have the same dimension, the row space and column space of a matrix need not be equal. Are they ever equal?

Show that the set $\{\hskip 1.00006pt(1,-1,2,-3),(1,1,2,0),(3,-1,6,-6)\hskip 1.00006pt\}$ does not have the same span as $\{\hskip 1.00006pt(1,0,1,0),(0,2,0,3)\hskip 1.00006pt\}$. What, by the way, is the vector space?

Show that this set of column vectors

is a subspace of $\mathbb{R}^{3}$. Find a basis.

Show that the transpose operation is linear:

for $r,s\in\mathbb{R}$ and $A,B\in\mathcal{M}_{m\!\times\!n}$.

In this subsection we have shown that Gaussian reduction finds a basis for the row space.

• Show that this basis is not unique — different reductions may yield different bases.

• Produce matrices with equal row spaces but unequal numbers of rows.

• Prove that two matrices have equal row spaces if and only if after Gauss-Jordan reduction they have the same nonzero rows.

Why is there not a problem with Remark 3.15 in the case that $r$ is bigger than $n$?

Show that the row rank of an $m\!\times\!n$ matrix is at most $m$. Is there a better bound?

Show that the rank of a matrix equals the rank of its transpose.

True or false: the column space of a matrix equals the row space of its transpose.

We have seen that a row operation may change the column space. Must it?

Prove that a linear system has a solution if and only if that system’s matrix of coefficients has the same rank as its augmented matrix.

An $m\!\times\!n$ matrix has full row rank if its row rank is $m$, and it has full column rank if its column rank is $n$.

• Show that a matrix can have both full row rank and full column rank only if it is square.

• Prove that the linear system with matrix of coefficients $A$ has a solution for any $d_{1}$, …, $d_{n}$’s on the right side if and only if $A$ has full row rank.

• Prove that a homogeneous system has a unique solution…

How would the conclusion of Lemma 3.3 change if Gauss’s Method were changed to allow multiplying a row by zero?

What is the relationship between $\operatorname{rank}(A)$ and $\operatorname{rank}(-A)$? Between $\operatorname{rank}(A)$ and $\operatorname{rank}(kA)$? What, if any, is the relationship between $\operatorname{rank}(A)$, $\operatorname{rank}(B)$, and $\operatorname{rank}(A+B)$?