Linear algebra help.. FORM basis of R3

Here is the problem:-
http://s2.postimage.org/ml73w3leq/Vector…

Can someone help me out with that problem?
Thanks a lot

So what we want to know is whether these vectors span R^3. In other words, can any vector in R^3 be written as au + bv + cw where a,b,c are real numbers and u,v,w are your three vectors. If the only way for au + bv + cw to equal 0 is for a,b,c to all equal 0, then they are linearly independent, and thus form a basis.

This amounts to solving the system of equations based on these three vectors. We have

au1 + bv1 + cw1 = 0
au2 + bv2 + cw2 = 0
au3 + bv3 + cw3 = 0

where u1 represents the first component of u1, etc.

So make the matrix with your column vectors u,v,w.

2...1...4
1...0...3
-1..1...2

Row reduce this.

2...1...4
-1..1...2
1...0...3


2...1...4
0...1...5
1...0...3


2...1...4
0...1...5
0..-1...2


1...0...0
0...1...0
0...0...1

So we see that the three are linearly independent and thus form a basis.