Find bases over the following subspaces of R3 A x y zT 2x

Find bases over the following subspaces of R^3. A = {(x, y, z)^T : 2x + y - z = 0}. B = {(x, y, z)^T : x + y -2x = 0, x - y = 0,}

Solution

(a) we have 2x+y-z=0

or z = 2x+y

so for vector x for the above , vector x = ( x, y , 2x+y)

= x ( 1, 0 , 2) + y( 0, 1, 1)

So bases can be { { 1,0,2) , (0,1,1) }

(b) x+y - 2z =0

or z = (x+y)/2

so for vector x for the above ,

vector x = (x , y, (x+y)/2)

   = x (1, 0, 1/2) + y( 0, 1, 1/2)

So bases can be chosen as {(1,0, 1/2) , (0, 1, 1/2)}