Solve Vector v as Linear Combination of x & y

[mr_coffee]

Okay this problem seems easy, and i t hought i understood how to do it, but no.

Express the vector v= [14 -12]^T as a linear combination of:
x = [2 -1]^T; y = [-1 3];
v = _x + _y;
_ means that's were they want me to put an answer.

So i row reduced
2 -1
-1 3
which is
1 0
0 1

so why wouldn't the answer just be
v = 14x -12y

because
14 * [1 0] = [14 0];
-12*[0 1] = [0 -12];
which is [14 -12] if u add them so what the?

 

[HallsofIvy]

Because x and y are NOT [1, 0] and [0,1]! Saying that the "row-reduce" to [1,0] and [0,1] does not mean they are equal to them.

I would suggest not row reducing at all:
$$\alpha x+ \beta y= [2\alpha, -\alpha]+ [-\beta,3\beta]$$
$$= [2\alpha- \beta,-\alpha+ 3\beta]= [14, -12]$$
so you must have $2\alpha- \beta= 14$ and $-\alpha+ 3\beta= -12$. Can you solve those two equations?

 

[mr_coffee]

Ahhh, i c, thank you Ivy! It worked out great!