Question

Find a basis B for the domain of T such that the matrix for T relative to B is diagonal.

T: R3 → R3: T(x, y, z) = (−4x + 2y − 3z, 2x − y − 6z, −x − 2y − 2z)

B =

Answer 1

T: R³ → R³ is defined by T(x, y, z) = (−4x + 2y − 3z, 2x − y − 6z, −x − 2y − 2z)

If 2x − y − 6z = 0 and −x − 2y − 2z = 0, then x = 2z and y =-2z so that (x,y,z) = (2z,-2z,z) = z(2,-2,1).

If −4x + 2y − 3z = 0 and −x − 2y − 2z = 0, then x = -z and y = -z/2 so that (x,y,z) = ( -z,-z/2,z) = (z/2)( -2,-1,2).

If −4x + 2y − 3z =0 and 2x − y − 6z = 0, then x = y/2 and z= 0 so that (x,y,z) = (y/2,y,0) = (y/2)(1,2,0).

Now, let B = { (2,-2,1),(-2,-1,2),(1,2,0)}.

Then T(2,-2,1) = (-15,0,0), T(-2,-1,2) = ( 0,-15,0) and T(1,2,0) = (0,0,-5) so that [T]_B =

-15	0	0
0	-15	0
0	0	-5

It may be observed that [T]_B is a diagonal matrix.