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Exercise 3.C.4

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Proof. If $T v_{1} = 0$ , then all entries in the first column is zero for any basis of $W$ . If $T v_{1} \neq 0$ , let $w_{1} = T v_{1}$ and extend it a basis $w_{1}, \dots, w_{n}$ of $W$ . Then

T v_{1} = 1 w_{1} + 0 w_{2} + \dots + 0 w_{n},

which shows that the entries in the first column are all $0$ except the first one, which is $1$ . □

celiopassos

2017-10-06 00:00

Exercise 3.C.4

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