Can Linear Algebra Prove Vector Dependencies and Transformations?

[mafquestion]

1) prove that for any five vectors (x1, ..., x5) in R3 there exist real numbers (c1, ..., c5), not all zero, so that BOTH

c1x1+c2x2+c3x3+c4x4+c5x5=0 AND c1+c2+c3+c4+c5=0

2)Let T:R5-->R5 be a linear transformation and x1, x2 & x3 be three non-zero vectors in R5 so that
T(x1)=x1
T(x2)=x1+x2
T(x3)=x2+x3

prove that {x1, x2, x3} are three linearly independent vectors.

any help would be greatly appreciated, thank you!

 

[aPhilosopher]

I've thought up a proof for the first one but it might be too complicated. I'll try to think of a simpler one if somebody else doesn't.

As for the second, assume that you have a linear combination of the three equal to zero. Map it under the matrix and see if something cool happens. Then see if it happens again. There's probably a contradiction with the assumptions in there somewhere ;)

 

[Office_Shredder]

For question 1)

Extend a vector in R³ to one in R⁴ by adding a 1 in the fourth entry.