Find a basis for the subspace S of R4 consisting of all vect

Find a basis for the subspace S of R^4 consisting of all vectors of the form (a - 2b + c,b - c,c,b + a)^T, where a, b, and c range over all real numbers. What is the dimension of S

Solution

Just factor out a, b, c:

( a -2b+c , b-c , c , b +a )^T = a ( 1 , 0 , 0 , 1)^T + b( -2, 1 , 0 , 1)^T + c( 1 , -1, 1 , 0 )^T

These vectors form the basis < 1, 0, 0 , 1> , < -2 , 1 , 0 , > , < 1 , -1 , 1 , 0 >

Dimension is 3