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## Homework Statement

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**=

*c1*

**v1**+

*c2*

**v2**. (Hint: Express

**w**in component form and obtain two linear equations for the unknowns

*c1*and

*c2*.

## Homework Equations

## 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**=

*c1*

**v1**+

*c2*

**v2**, 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:

*c1*

**v1**+

*c2*

**v2**=

**w**= <

*c1 – c2*,

*c1 + c2*> = <0, 0>

=>

*c1 – c2*= 0,

*c1*=

*c2*

and

*c1 + c2*= 0

=>

*c1*+

*c1*= 2

*c1*= 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. :(