Let W span 1 2 1 0 2 1 1 1 Find a basis of W SolutionWrt t

Let W = span {[1 2 1 0], [2 1 1 1 ]}. Find a basis of W^.

Solution

Wrt the standard inner product , a vector [x y z w]\' is in W* (we use this notation for W perp)

iff       x+2y+z=0

and   2x+y+z+w=0

Subtracting , we get

                  w = y-x

and z = -2y-x.

Clearly W* is a 2-dimensional subspace and a basis is given by setting x=1,y=0 for the first vector and x=0 and y=1 for the second vector.

So a basis for W* is [1 0 -1 -1]\' and [0 1 -2 1]\' ( where \' stands for the transpose)