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

Trouble w/ Lemma

  1. Oct 6, 2004 #1
    The text I am using has proved the following thereom near the beginning of the chapter: If two vector spaces V, W are equidimensional (finite) and T is a linear transformation from V to W, then one-to-one and onto are equivalent. It has also used the result liberally in latter sections.

    Trouble oocurs when it comes to a lemma near the end of the chapter. The text suddenly seems to "forget" the preceding thereom. The lemma is: "Let V be a vector space, and suppose that T and U are linear operators [transformations onto same vector space] on V such that U is onto and the null spaces of T and U are finite-dimensional. Then the null space of TU is finite-dimensional, and nullity TU = nullity T + nullity U." This is followed by a lengthy proof. But, according the the previous result, U is also one-to-one. This in turn readily means the nullity U is always zero. I don't understand why this is wholly ignored.
  2. jcsd
  3. Oct 6, 2004 #2
    oh.. nm. The lemma never states V is finite dimensional.
  4. Oct 7, 2004 #3


    User Avatar
    Homework Helper
    Gold Member

    What does Lemma mean?
  5. Oct 7, 2004 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Auxiliary proposition.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Trouble w/ Lemma
  1. Yoneda lemma (Replies: 3)

  2. Urysohn lemma? (Replies: 14)

  3. Interesting lemma (Replies: 15)

  4. Lemma Lucas theorem (Replies: 6)