Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Linear Transformation questions about dimensions

  1. Feb 8, 2012 #1
    1. Say you have a linear transform from A to B, and where A has a higher dimension than B. How do you show that the kernel of the transform has more than one element (i.e. 0)? Also, if B has a higher dimension than A, then how to show that the transform isn't surjective?

    2. The attempt at a solution

    By showing that the kernel has more than the element 0, I want to show that the transform isn't injective. But I'm not quite sure how to get there just by using the fact that A has a higher dimension than B. Is that a good way(as in, not too complicated) of proving it? Any ideas?

    For the other part, it makes sense intuitively, since the basis of A will have less elements than the basis of B, so there shouldn't be a surjection. But how do you proceed from there to show that the image of A is a proper subset of B?
  2. jcsd
  3. Feb 8, 2012 #2
    Think of the theorem that says the dim(A)=dim(Ker(f)) + dim(Im(f)), where f:A->B is a linear morphism (transformation)
  4. Feb 8, 2012 #3
    Thanks! Now I feel really stupid for not considering Rank-Nullity before asking this....
  5. Feb 8, 2012 #4
    You're welcome. I feel stupid for not knowing that theorem has a name ...
  6. Feb 8, 2012 #5
    Do you know any bad math jokes related to this?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook