MHB Understanding Null Space and Column Space of a Matrix

[karush]

$\tiny{311.q.02.05}\\$
Let
$A=\begin{bmatrix}
1 & 5 & -4 & -3 & 1 \\
0 & 1 & -2 & 1 & 0 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix},
u=\begin{bmatrix}
7\\ 0\\ 1 \\ 2 \\ 3
\end{bmatrix}
and \,
v=\begin{bmatrix} 3\\ 5\\ 0 \end{bmatrix}$
$\textsf{(a) Find Nul} \textbf{A} $
$\textsf{(b) Find Col} \textbf{A} $ok basically clueless, wasn't there for lecture

 

[S.G. Janssens]

You probably have a book or lecture notes?
What are the definitions of $\operatorname{Nul}{A}$ and $\operatorname{Col}{A}$ they give there? (These definitions should not involve $u$ nor $v$.)

 

[karush]

You probably have a book or lecture notes?
What are the definitions of $\operatorname{Nul}{A}$ and $\operatorname{Col}{A}$ they give there? (These definitions should not involve $u$ nor $v$.)

this was on a practice quiz
not from the text. this is all that was said.

 

[S.G. Janssens]

this was on a practice quiz
not from the text. this is all that was said.

That seems a little strange. There must be a place where your teacher (or the book he is using) has defined $\operatorname{Nul}{A}$ and $\operatorname{Col}{A}$. I strongly recommend that you look it up yourself and compare with the following:

$\operatorname{Nul}{A}$ abbreviates the nullspace of $A$. Assuming you work with real numbers, for this particular matrix it is the set of all vectors $\mathbf{x}$ in $\mathbb{R}^5$ such that $A\mathbf{x} = 0$. So, finding $\operatorname{Nul}{A}$ is equivalent to finding the solution space of the homogeneous system corresponding to $A$. Do you understand this?

On the other hand, $\operatorname{Col}{A}$ abbreviates the column space of $A$. In this example it is the set of all $\mathbf{y} \in \mathbb{R}^3$ such that $A\mathbf{x} = \mathbf{y}$ for some $\mathbf{x} \in \mathbb{R}^5$. So, $\operatorname{Col}{A}$ is the linear span of the columns of $A$. Are all the columns needed? How can you find out which ones are? (For this, note that $A$ is already in row echelon form.)

 

[S.G. Janssens]

I'm sorry for answering. Indeed, if the OP does not want to make any effort, then why should I?

Please just look up what $\operatorname{Nul}{A}$ and $\operatorname{Col}{A}$ mean, make sure that you understand the definition precisely, and do the exercise.