linear algebra Show that the vectors a 1 1 0 b 0 1 2 c 1

linear algebra

Show that the vectors a- = [1, -1, 0], b- = [0, 1, 2], c- = [1, 0, -1] form a basis and find the coordinates of d- = [2, 3, 0] in this basis 20%

Solution

Let, r,s,t so that

ra+sb+tc=0

This gives us

r+t=0

-r+s=0

2t-t=0

Hence, r=s=t=0

SO vectors,a,b,c are linearly independent

And hence form a basis for R3 because R3 has dimension 3 and hence any linearly independent set with 3 vectors forms a basis.

Let, A=[2 3 0]=ra +sb+tc

2=r+t

3=-r+s

0=r

HEnce, t=2,s=3

So, A=3b+2c