Question

Let T be a normal linear operator on a finite-dimensional complex inner product space V. Let W be a subspace of V. Prove that W is a T-invariant subspace if and only if W is a T*-invariant subspace.

Answer 1

To prove that a subspace W of a finite-dimensional complex inner product space V is T-invariant if and only if it is T*-invariant, we can use the properties of inner product spaces and the adjoint operator.

Step-by-step explanation:

To prove the given statement, we need to show that a subspace W of a finite-dimensional complex inner product space V is T-invariant if and only if it is T*-invariant. Let's start with the forward direction. Assume W is T-invariant. This means that for any vector w in W, T(w) is also in W. Now, let's consider the adjoint, T*, of the linear operator T. Since V is a finite-dimensional complex inner product space, T* exists. We want to show that for any vector w in W, T*(w) is also in W. To do this, we can take the inner product of T*(w) with any vector v in V and use the properties of inner product spaces.

Now let's move on to the reverse direction. Assume W is T*-invariant. This means that for any vector w in W, T*(w) is also in W. We want to show that for any vector w in W, T(w) is also in W. Similar to the forward direction, we can take the inner product of T(w) with any vector v in V and use the properties of inner product spaces to prove this statement.