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Kreizhn
#1
Mar10-11, 03:10 PM
P: 743
1. The problem statement, all variables and given/known data
Here is a really simple lin.alg problem that for some reason I'm having trouble doing.

Assume that [itex]\left\{ v_i \right\} [/itex] is a set of linearly independent vectors. Take w to be a non-zero vector that can be written as a linear combination of the [itex] v_i [/itex]. Show that [itex] \left\{ v_i - w \right\} [/itex] is still linearly independent.

3. The attempt at a solution
For some reason I'm quite stuck on this. My first goal was to let [itex] b_i [/itex] be such that we can write
[tex] w = \sum_j b_j v_j [/tex]
and then consider the sum
[tex] \sum_i a_i (v_i-w) = 0 [/itex]
and show that each [itex] a_i [/itex] must necessarily be zero. Substituting the first equation into the other yields
[tex] \begin{align*}\sum_i a_i (v_i - \sum_j b_j v_j ) &= \sum_i a_i - \sum_{i,j} a_i b_j v_j \\
&= \sum_i \left( a_i - \sum_j a_j b_i \right) v_i
\end{align*}[/tex]
where in the last step I've switched the indices in the double summation. By linear independence of the [itex] v_i [/itex] it follows that
[tex] a_i = \sum_j a_j b_i [/tex]
but that's where I'm stuck.

It's possible that I'm doing this the wrong way also. Any help would be appreciated.
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