Linear maps (rank and nullity)

[NewtonianAlch]

Homework Statement

A linear map T: ℝ^{m}-> R^{n} has rank k. State the value of the nullity of T.

The Attempt at a Solution

I know that rank would be the number of leading columns in a reduced form and the nullity would simply be the number of non-leading columns, or total columns - rank.

As we are not given how many different vectors are in the matrix, I'm not too sure how to give a value.

 

[CompuChip]

nullity would simply be [...] total columns - rank.

The rank is given.
What is the total number of columns for a map \mathbb{R}^m \to \mathbb{R}^n?

 

[NewtonianAlch]

I'm inclined to say m, but I'm not entirely sure. So m - k? Considering that's the starting space if you will.

 

[HallsofIvy]

The "rank-nullity theorem" which, if you were given this problem, you are probably expected to know, says that if A is a linear transformation from a vector space of degree m to a vectoer space of degree n then rank(A)+ nullity(A)= m.

 

[CompuChip]

I'm inclined to say m, but I'm not entirely sure. So m - k? Considering that's the starting space if you will.

If it's a map from \mathbb{R}^m to \mathbb{R}^n, that means that you should be able to multiply the matrix with a vector with m components, and this should give a vector with n components. If you think about the way matrix multiplication works, does the number of columns need to be m or n?