Find a basis for as well as the dimension of the subspace th

Find a basis for, as well as the dimension of, the subspace: the subspace of R^5 of all the vectors with the first and second componenets zero and the last component equal to the sum of the first four.

Solution

The subspace consists of vectors
{(x1, x2, x3, x4,x5) R^5

given first and second components are zero
x1=0, x2=0

the last component equal to the sum of the first four.
x5 = x1+x2+x3+x4
x5 = 0 +0 +x3 +x4
x3 +x4 =x5

{ x1=0, x2=0, x3 +x4 =x5}.

It is the nullspace of the matrix
1 0 0 0 0
0 1 0 0 0
0 0 1 1 -1

, a two dimensional subspace of R^5
,
so any two independent vector gives a basis. For example, we can take as a basis
v1 = (0, 0, 1, 1,2), v2 = (0, 0, 4,-3,1).