View Single Post

## Linearly Independent Sets After Subtraction

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 $\left\{ v_i \right\}$ 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 $v_i$. Show that $\left\{ v_i - w \right\}$ 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 $b_i$ be such that we can write
$$w = \sum_j b_j v_j$$
and then consider the sum
\sum_i a_i (v_i-w) = 0 [/itex] and show that each $a_i$ 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*}
where in the last step I've switched the indices in the double summation. By linear independence of the $v_i$ it follows that
$$a_i = \sum_j a_j b_i$$
but that's where I'm stuck.

It's possible that I'm doing this the wrong way also. Any help would be appreciated.
 PhysOrg.com science news on PhysOrg.com >> New language discovery reveals linguistic insights>> US official: Solar plane to help ground energy use (Update)>> Four microphones, computer algorithm enough to produce 3-D model of simple, convex room