4. Suppose that T is a normal operator on V and that 3 and 4 are eigenvalues of T. Prove that there exists a vector v epsilon V such that ||v|| = root 2 and ||T(v)|| = 5. HINT: Think about what we know about the eigenspaces of normal opera tors, then create a vector from the two eigenspaces and use the Pythagorean theorem to verify the vector has the desired properties. Solution Given that 3 and 4 are eigen values of T. Then there exists a vector v such that v = xi+yj such that |x|=1 and |y| =1 as T is normal Hence ||V|| = rt of x^2+y^2 = rt 2 ||T(v)|| = rt of 3^2+4^2 = 5.

1. 4. Suppose that T is a normal operator on V and that 3 and 4 are eigenvalues of T. Prove that
there exists a vector v epsilon V such that ||v|| = root 2 and ||T(v)|| = 5. HINT: Think about what
we know about the eigenspaces of normal opera tors, then create a vector from the two
eigenspaces and use the Pythagorean theorem to verify the vector has the desired properties.
Solution
Given that 3 and 4 are eigen values of T.
Then there exists a vector v such that v = xi+yj such that |x|=1 and |y| =1 as T is normal
Hence ||V|| = rt of x^2+y^2 = rt 2
||T(v)|| = rt of 3^2+4^2 = 5