genxhis
Oct6-04, 12:18 AM
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.
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.