Understanding Null Space and Column Space of a Matrix

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  • Thread starter karush
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In summary: There is probably a place in your textbook or lecture notes where these definitions are given. $\operatorname{Nul}{A}$ is the set of all vectors in $\mathbb{R}^5$ such that $A\mathbf{x} = 0$, and $\operatorname{Col}{A}$ is the set of all vectors in $\mathbb{R}^3$ such that $A\mathbf{x} = \mathbf{y}$.
  • #1
karush
Gold Member
MHB
3,269
5
$\tiny{311.q.02.05}\\$
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
 
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  • #2
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 $v$.)
 
  • #3
Krylov said:
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.
 
  • #4
karush said:
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.)
 
  • #5
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.
 

FAQ: Understanding Null Space and Column Space of a Matrix

1. What does "Find Nul A Find Col A (q2)" mean?

"Find Nul A Find Col A (q2)" is a command used in linear algebra to find the null space and column space of a matrix A.

2. How do I use "Find Nul A Find Col A (q2)" in my research?

"Find Nul A Find Col A (q2)" can be used to solve systems of linear equations, determine the rank of a matrix, and find the basis of a vector space, all of which are important in various scientific fields such as physics, engineering, and economics.

3. Can "Find Nul A Find Col A (q2)" be used for non-square matrices?

Yes, "Find Nul A Find Col A (q2)" can be used for non-square matrices. However, the dimensions of the null space and column space will differ depending on the dimensions of the matrix.

4. Are there any limitations to using "Find Nul A Find Col A (q2)"?

One limitation of using "Find Nul A Find Col A (q2)" is that it can only be used for numerical matrices, not symbolic ones. Additionally, it may not be applicable for very large matrices due to computational limitations.

5. How does "Find Nul A Find Col A (q2)" relate to other linear algebra concepts?

"Find Nul A Find Col A (q2)" is related to other linear algebra concepts such as linear independence, basis, and dimension. It also has applications in eigenvalues and eigenvectors, matrix transformations, and solving linear systems using methods such as Gaussian elimination and matrix inversion.

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