Question

Express each of the standard basis vectors as linear combinations

Answer 1

standard basis vectors are e1=(1,0,0), e2=(0,1,0) and e3=(0,0,1).

To express each of the standard basis vectors as linear combinations of the vectors in B, b1=(1,0,0), b2=(−1,1,0) and b3=(0,1,1) you want to calculate scalar constants uk, vk and wk such that

e1=u1b1+v1b2+w1b3

e2=u2b1+v2b2+w2b3

e3=u3b1+v3b2+w3b3

You can represent the above system of equations in matrix form (forming a matrix whose columns are b1, b2 and b3, so that),

⎛⎝⎜100−110011⎞⎠⎟⎛⎝⎜ukvkwk⎞⎠⎟=ek

In terms of how to calculate the scalar constants, you can use a variety of means, and I don't think you have to use Gaussian elimination - using simultaneous equations is just as fine.

For e1=(1,0,0), note that b1=e1, which is quite simple, so you obtain u1=1, v1=0 and w1=0. So e1=b1.

For e2=(0,1,0), we have u2−v2=0, v2+w2=1,w2=0, so that u2=1,v2=1 and w2=0. So e2=b1+b2.

For e3=(0,0,1), we have u2−v2=0, v2+w2=0,w2=1, so that u2=−1,v2=−1 and w2=1.So e3=−b1−b2+b3.