Trouble w/ Lemma

  • Thread starter genxhis
  • Start date
  • #1
37
1

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
37
1
oh.. nm. The lemma never states V is finite dimensional.
 
  • #3
JasonRox
Homework Helper
Gold Member
2,314
3
What does Lemma mean?
 
  • #4
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
JasonRox said:
What does Lemma mean?
Auxiliary proposition.
 

Related Threads on Trouble w/ Lemma

Replies
9
Views
3K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
14
Views
3K
  • Last Post
Replies
15
Views
4K
  • Last Post
Replies
3
Views
696
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
5
Views
4K
  • Last Post
Replies
3
Views
2K
Replies
4
Views
957
Top