Exercise 2.6.1

Answers

1.: No. Every linear functional is a linear transformation.
2.: Yes. It’s domain and codomain has dimension $1$ .
3.: Yes. They have the same dimension.
4.: Yes. It’s isomorphic to the dual space of its dual space. But if the “is” here in this question means “equal”, then it may not be true since dual space must has that its codomain should be $𝔽$ .
5.: No. For an easy example we may let $T$ be the linear transformation such that $T (x_{i}) = 2 f_{i}$ , where $β {x_{1}, x_{2}, \dots, x_{n}}$ is the basis for $V$ and $β^{∗} {f_{1}, f_{2}, \dots, f_{n}}$ is the corresponding dual basis for $V^{∗}$ .
6.: Yes.
7.: Yes. They have the same dimension.
8.: No. Codomain of a linear functional should be the field.

jephianlin

2011-06-27 00:00