Let v1 = <1, 1> and v2 = <-1, 1>. Show that for any vector w in the plane one can find constants c1 and c2 so that w = c1v1 + c2v2. (Hint: Express w in component form and obtain two linear equations for the unknowns c1 and c2.
The Attempt at a Solution
Alright, so the geometric proof to this question is easy enough, but our professor wants us to find an algebraic one, and he said that it's a good example of an exam question. So I want to make sure I understand the proofs behind these questions before exam time comes.
Since w = c1v1 + c2v2, first I multiplied c1 and c2 by the vectors <1, 1> and <-1, 1> respectively:
w = c1<1, 1> + c2<-1, 1>
w = <c1, c1> + <–c2, c2>
w = <c1 – c2, c1 + c2>
Hence, I have the vector w in terms of its components. And this is where I'm stuck. I looked online at stuff about the linear dependence of vectors, and how if you equate the vector set to 0 and that is the only solution then they are linearly independent and that you can't form the third vector w from them. And in fact that's what happened when I tried to go further with that:
c1v1 + c2v2 = w = <c1 – c2, c1 + c2> = <0, 0>
=> c1 – c2 = 0, c1 = c2
and c1 + c2 = 0
=> c1 + c1 = 2c1 = 0, in which case I just get c1 = 0 and then hence c2 = 0, and since 0 is the only solution I get that they are linearly independent even though I *know* for sure that geometrically you can make them into w.
Help would be much appreciated! I've been stuck on this for a while and it's really starting to irritate me. :(