Trouble w/ Lemma: Nullity U Ignored in Finite Vector Space

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Discussion Overview

The discussion revolves around a lemma related to linear transformations in vector spaces, specifically addressing the implications of nullity in the context of finite-dimensional vector spaces. Participants explore the relationship between the lemma and a previously established theorem regarding the equivalence of one-to-one and onto transformations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes a theorem stating that if two finite vector spaces are equidimensional and a linear transformation is applied, then being one-to-one and onto are equivalent.
  • The same participant expresses confusion about a lemma that seems to contradict this theorem by stating that the nullity of a transformation TU is the sum of the nullities of T and U, despite U being onto, which implies its nullity should be zero.
  • Another participant points out that the lemma does not specify that V is finite-dimensional, which may affect the interpretation of the theorem and lemma.
  • Two participants inquire about the meaning of the term "lemma," with one providing a definition as an auxiliary proposition.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the implications of the lemma in relation to the theorem. There is acknowledgment of a potential oversight regarding the dimensionality of V, but no agreement on the resolution of the confusion presented.

Contextual Notes

The discussion highlights the importance of dimensionality in understanding the relationship between the theorem and the lemma, suggesting that assumptions about the dimensionality of vector spaces are critical to the claims being made.

genxhis
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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.
 
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oh.. nm. The lemma never states V is finite dimensional.
 
What does Lemma mean?
 
JasonRox said:
What does Lemma mean?
Auxiliary proposition.
 

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