Linear maps (rank and nullity)

  • #1

Homework Statement


A linear map T: ℝ[itex]^{m}[/itex]-> R[itex]^{n}[/itex] 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.
 

Answers and Replies

  • #2
CompuChip
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nullity would simply be [...] total columns - rank.

The rank is given.
What is the total number of columns for a map [itex]\mathbb{R}^m \to \mathbb{R}^n[/itex]?
 
  • #3
I'm inclined to say m, but I'm not entirely sure. So m - k? Considering that's the starting space if you will.
 
  • #4
HallsofIvy
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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.
 
  • #5
CompuChip
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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 [itex]\mathbb{R}^m[/itex] to [itex]\mathbb{R}^n[/itex], 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?
 

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