Linear Algebra Find a basis for the subspace of R4 that is s

Linear Algebra

Find a basis for the subspace of R^4 that is spanned by the vectors v1 = (1, 0, 0), v2 = (1, 0, 1) v3 = (2, 0, 1), v4 = (0, 0, -1) Be sure to explain why what you claim to be a basis is indeed a basis.

Solution

To find a basis for v1, v2, v3, v4

Any vector in R^2 is of the form (x,y,z,w)

If it is represented as a linear combination of vis

then (x,y,z,w) = av1+bv2+cv3+dv4

i.e. x = a+b+2c

y = 0

z= b+c-d

But this shows that y is always 0

In R^4 of (x,y,z,w) y is always 0 shows that it is actually R3 with variables only 3.

Hence cannot find a basis.