Question 4
Let V be a finite dimensional complex inner product space of dimension n ? 3, and let T:
V ? V be a linear operator. Suppose that W is a subspace of V with orthonormal basis
{e?, e?, ..., e?}, where 1 ? k < n, with the property that ||T(w)|| = ||w|| for all w ? W and
T(x) = 0 for all x ? W?, where W? denotes the orthogonal complement of W.
(a) Use the Polarization Identity
$\qquad (u, v) = \frac{1}{4}(||u + v||² - ||u - v||² + i||u + iv||² - i||u - iv||²)$
to show that $(T(x), T(y)) = (x, y)$ for all x, y ? W.
(3 marks)
(b) Show that {T(e?), T(e?), ..., T(e?)} is an orthonormal basis of Range(T), where
Range(T) = {T(x) : x ? V}.
(4 marks)
(c) Let T* denote the adjoint of T and let U = Range(T).
Prove the following:
T*(T(e?)) = e? for all i, 1 ? i ? k, ||T*(u)|| = ||u|| for all u ? U, and T*(x) = 0 for
all x ? U?.
(10 marks)
(d) Give an example to show that W need not be a T-invariant subspace.
(3 marks)
