What is the dimension of the subspace W = {(x, y, z,w) 2 R4 | x + y + z = 0} of R4?
Find a basis for this subspace.
Solution. The subspace is defined as the solution space of the single equation x +
y + z = 0 in the 4 variables x, y, z,w. Thus there should be 3 free variables, and the
subspace will have dimension 3. We choose y, z,w for the free variables–they can
take
on any values in R–and then the value of x will be determined by x = −y − z. Hence
we can express the given subspace as
{(−y−z, y, z,w)|y, z,w 2 R} = {y(−1, 1, 0, 0)+z(−1, 0, 1, 0)+w(0, 0, 0, 1)|y, z,w 2 R},
which equals span{(−1, 1, 0, 0), (−1, 0, 1, 0), (0, 0, 0, 1)}. Since these 3 vectors are
linearly
independent, they form a basis for W, and dimW = 3.