How to correctly solve this problem? (linear dependency)

[BobJimbo]

This is the problem:

Suppose a, b and c are linearly independent vectors. Determine whether or not the
vectors (a + b), (a - b), and (a - 2b + c) are linearly independent.

Here was my solution, which involved writing words (and hasn't actually been confirmed correct yet):

Let's align a, b and c with the x-y-z axis so that a and b have only x and y components, whereas c has z too. This is true for any set of linearly independent vectors. This would mean no combination of (a + b) and (a - b) can create (a - 2b + c), because neither has a z component, and therefore (a + b), (a - b), and (a - 2b + c) are also linearly independent.

I'd like to know what the most efficient solution is, which I assume is algebraic and doesn't involve so many words. Thanks!

 

[Buzz Bloom]

Hi Bob:

Your approach is is good. Each of the three combination vectors can be expressed as a triple of (x,y,z) values. It is possible to chose the scale of the x, y, and z coordinates such that a=b=c=1 If you form a matrix with these three triples, the determinant will equal zero if the three triples are not independent. Note that this 3x3 matrix will have elements that are all simple integers.

Hope this helps.

Regards,
Buzz