# Homework Help: Question about row/column/nullspace

1. Apr 12, 2017

### Arnoldjavs3

1. The problem statement, all variables and given/known data
If we have a 3x5 matrix:

The row space is in r5, the col space is in r3, and the nullspace is in r3 correct?
Because you would need 5 components to be a member of r5 so the col space cannot be a member of r5 correct?

Here is the question: http://prntscr.com/evo91g

2. Relevant equations

3. The attempt at a solution
Am I getting something fundamentally wrong here? Or is the question just wrong? I believe I'm right for reasons mentioned above.

2. Apr 12, 2017

### Math_QED

The nullspace is a subspace of $\mathbb{R}^5$, right? Therefore the nullspace is not in $\mathbb{R}^3$. For the rest you seem correct. It seems the question is wrong.

3. Apr 12, 2017

### Arnoldjavs3

Why would it be in r5 though? Each vector in the nullspace will only have 3 components

4. Apr 12, 2017

### Math_QED

Can you give me your definition of null space?

5. Apr 12, 2017

### Arnoldjavs3

The subspace of linear combinations that make your system equal to 0?

edit: Now that I think about it, when A is a 3x5 matrix, you need a 5x1 matrix vector X. Since X is 5x1, that means the nullspace is in r5 since X represents all vectors inside the nullspace?

6. Apr 12, 2017

The null space and the row space of a matrix will always be sub-spaces of the same vector space (why?).The column space and row space of a matrix will be sub-spaces of the vector space whose dimension is the number of elements in the vector. So if we have a 12x23 matrix, its row space is a sub-space of R23 and its column space is a sub-space of R12. From here I assume you can figure out the correct solution.

7. Apr 12, 2017

### Arnoldjavs3

Yup! Makes sense now. Just find it interesting that I understand the nullspace and can compute it easily but I still abstract details like that.

8. Apr 13, 2017

### Math_QED

Yes, your reasoning is correct. That's a very bad definition of null space, though.

A better definition would be: Let $A$ be an $m \times n$ matrix.

$Null(A) := \{x \in \mathbb{R}^n| Ax = 0\}$

Or if you are familiar with linear mappings: Let $f: V \rightarrow W$ be a linear mapping:

$Ker(f) := \{x \in V|f(x) = 0\}$

Last edited: Apr 13, 2017