# Linear maps (rank and nullity)

## 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
Homework Helper
nullity would simply be [...] total columns - rank.

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

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
Homework Helper
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
Homework Helper
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?