Exercise 2.4.1

Answers

1.: No. It should be ${({[T]}_{α}^{β})}^{- 1} = {[T^{- 1}]}_{β}^{α}$ .
2.: Yes. See Appendix B.
3.: No. $L_{A}$ can only map $𝔽^{n}$ to $𝔽^{m}$ .
4.: No. It isomorphic to $𝔽^{5}$ .
5.: Yes. This is because $P_{n} (𝔽) ≅ 𝔽^{n}$ .
6.: No. We have that $(\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) (\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{matrix}) = I$ but $A$ and $B$ are not invertible since they are not square.
7.: Yes. Since we have both $A$ and ${(A^{- 1})}^{- 1}$ are the inverse of $A^{- 1}$ , by the uniqueness of inverse we can conclude that they are the same.
8.: Yes. We have that $L_{A^{- 1}}$ would be the inverse of $L_{A}$ .
9.: Yes. This is the definition.

jephianlin

2011-06-27 00:00