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Exercise 3.A.13

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Proof. If $v_{k} = 0$ for some $k$ when we can choose any nonzero vector as $w_{k}$ . So assume $v_{k} \neq 0$ for each $k$ . Let $n$ be the smallest integer such that $v_{1}, \dots, v_{n}$ is linearly indepedent. Then we can write

v_{n} = a_{1} v_{1} + \dots + a_{n - 1} v_{n - 1}

for some scalars $a_{1}, \dots, a_{n - 1} \in 𝔽$ . Then it suffices to choose any $w_{1}, \dots, w_{n}$ (the $w_{k}$ for $k > n$ won’t make a difference) such that

w_{n} \neq a_{1} w_{1} + \dots + a_{n - 1} w_{n - 1}

because if $T \in L (V, W)$ , with $T v_{k} = w_{k}$ for each $k$ , we must have

\begin{array}{l} w_{n} & = T v_{n} \\ = T (a_{1} v_{1} + \dots + a_{n - 1} v_{n - 1}) \\ = a_{1} T v_{1} + \dots + a_{n - 1} T v_{n - 1} \\ = a_{1} w_{1} + \dots + a_{n - 1} w_{n - 1} . \end{array}

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celiopassos

2017-10-06 00:00

Exercise 3.A.13

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