Question about row/column/nullspace

[Arnoldjavs3]

Homework Statement

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

Homework Equations

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.

 

[member 587159]

Homework Statement

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

Homework Equations

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.

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.

 

[Arnoldjavs3]

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.

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

 

[member 587159]

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

Can you give me your definition of null space?

 

[Arnoldjavs3]

Can you give me your definition of null space?

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?

 

[Adgorn]

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?

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.

 

[Arnoldjavs3]

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.

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.

 

[member 587159]

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?

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:

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