Homepage › Solution manuals › Stephen Friedberg › Linear Algebra › Exercise 2.1.33

Exercise 2.1.33

Answers

It’s natural that $R (T) \supset$ span $({T (v) : v \in β})$ since all $T (v)$ is in $R (T)$ . And for any $y \in R (T)$ we have $y = T (x)$ for some $x$ . But every $x$ is linear combination of finite many vectors in basis. That is, $x = \sum_{i = 1}^{k} a_{i} v_{i}$ for some $v_{i} \in β$ . So we have

y = T (\sum_{i = 1}^{k} a_{i} v_{i}) = \sum_{i = 1}^{k} a_{i} T (v_{i})

is an element in span $({T (v) : v \in β})$ .

jephianlin

2011-06-27 00:00

Exercise 2.1.33

Answers

Comments

Add answer