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Homework 5

(additional problems)

Problem 1   Let $ T:V\to V $ be a linear operator on a finite-dimensional vector space $ V $. Let $ \beta $ be a basis of $ V $ and $ A=[T]_{\beta } $. Suppose that $ P $ is an invertible  $ n\times n $ matrix. Show that there is a basis $ \gamma $ of $ V $ such that $ [T]_{\gamma }=PAP^{-1} $.

Problem 2   Let $ T:V\to W $ be a linear transformation.

a) (Rank-Nullity Theorem) If $ V $ is finite-dimensional, prove that $ \textrm{rank}(T)+\textrm{nullity}(T)=\textrm{dim}(V) $; (Give a direct proof, without using theorem of section 49 in the book)

b) Give an example of an infinite-dimensional vector space $ V $ and a linear operator $ T $ on $ V $ such that the statement of the rank-nullity theorem is false. Return to your proof of this theorem and think why it does not work in the infinite-dimensional case.

Problem 3   Prove that a linear transformation is one-to-one iff its null space consists of the zero vector only.

Problem 4   Prove that a linear transformation $ T:V\to W $, where $ V $ and $ W $ are finite-dimensional vector spaces with $ \textrm{dim}(V)=\textrm{dim}(W) $, is one-to-one if and only if $ \textrm{rank}(T)=\textrm{dim}(V) $. (Hint: first, prove that it is one-to-one iff it is onto; then prove that it is onto iff $ \textrm{rank}(T)=\textrm{dim}(V) $).




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Olga Radko 2003-08-04