Express as a linear combination

[fk378]

Homework Statement

The vectors v1= [1, -3], v2= [2, -8], v3= [ -3, 7] span R2 but do not form a basis. Find 2 different ways to express [1, 1] as a linear combination of v1, v2, v3.

The Attempt at a Solution

Since it states that the set is not a basis, then v3=v1+v2. I solved the system v1+v2=[1,1] and got 5v1-2v2=[1,1]. The second answer the book gives is 10v1-3v2+v3. How did they get the second answer?

 

[Dick]

If you solve c1*v1+c2*v2=[1,1] you get a unique solution since [v1,v2] are a basis. To get the second answer you solve c1*v1+c2*v2+c3*v3=[1,1]. Since [v1,v2,v3] are linearly dependent, and hence are not a basis, you get an infinite number of solutions. The books answer is only one of them.