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 $\textrm{rank}(T)=\textrm{dim}(V)$).