[Linear Algebra] Nullspace equals Column space

[Dafe]

Homework Statement

Why does no 3 by 3 matrix have a nullspace that equals its column space?

Homework Equations

NA

The Attempt at a Solution

$$A =<br /> \begin{bmatrix}<br /> 0 & 0 & 1 \\<br /> 0 & 0 & 0 \\<br /> 0 & 0 & 0<br /> \end{bmatrix}<br /> \]$$

$$C(A) =<br /> \begin{bmatrix}<br /> 1 \\<br /> 0 \\<br /> 0<br /> \end{bmatrix}<br /> \]$$

Does not then N(A) = C(A)?
I think I am missing something here.

Thank you for your time.

 

[HallsofIvy]

I don't know what you are missing because I don't know what you are thinking! The "column space" of a matrix is the space spanned by its columns thought of as vectors. The column space of your matrix is the one dimensional space spanned by <0, 0, 1>= $\vec{k}$. The null space of a matrix, A, is the set of all vectors, $\vec{v}$ such that $A\vec{v}= \vec{0}$. For this matrix that is the space spanned by <1, 0, 0>= $\vec{i}$ and <0, 1, 0>= $\vec{j}$. They are not at all the same. In fact the two are orthogonal complements.

It is true for any n by n matrix, with n odd, that the null space cannot be the same as the column space because, for any n by n matrix, the sum of the dimension of the column space and the dimension of the null space must equal n. If the two dimensions are the same, their sum is an even number.

 

[Dafe]

Hi,

I forgot the "space" in nullspace.

The book writes:

n - r = r

n = 3

3 = 2r is impossible.

The n - r = r is confusing me. Is r meant to be the number of pivot columns?

Thanks!

 

[Billy Bob]

Hi,

I forgot the "space" in nullspace.

The book writes:

n - r = r

n = 3

3 = 2r is impossible.

The n - r = r is confusing me. Is r meant to be the number of pivot columns?

Thanks!

You have probably learned a theorem like "rank + nullity = number of columns" or "rank + dimension of null space = number of columns." Yes, rank = number of pivot entries in rre form.

Also "rank= row rank = col rank = dim of col sp."

r = dim of col sp
n-r = dim of null sp

Set n-r equal to r; is the resulting equation possible?

 

[Dafe]

Ah yes, n - r is the number of special solutions, number of free variables and the dimension of the nullspace.
Thank you very much!