Page 253 - Linear Algebra Done Right

Chapter 10. Trace and Determinant
244
Exercises
1. Suppose T ∈L(V) and (v 1 ,...,v n ) is a basis of V. Prove that

M T, (v 1 ,...,v n ) is invertible if and only if T is invertible.
2. Prove that if A and B are square matrices of the same size and
AB = I, then BA = I.
3. Suppose T ∈L(V) has the same matrix with respect to every ba-
sis of V. Prove that T is a scalar multiple of the identity operator.

4. Suppose that (u 1 ,...,u n ) and (v 1 ,...,v n ) are bases of V. Let
T ∈L(V) be the operator such that Tv k = u k for k = 1,...,n.
Prove that

M T, (v 1 ,...,v n ) =M I, (u 1 ,...,u n ), (v 1 ,...,v n ) .

5. Prove that if B is a square matrix with complex entries, then there
exists an invertible square matrix A with complex entries such
that A −1 BA is an upper-triangular matrix.

6. Give an example of a real vector space V and T ∈L(V) such that
2
trace(T )< 0.
7. Suppose V is a real vector space, T ∈L(V), and V has a basis
2
consisting of eigenvectors of T. Prove that trace(T ) ≥ 0.
8. Suppose V is an inner-product space and v, w ∈L(V). Define
T ∈L(V) by Tu = u, v w. Find a formula for trace T.

2
9. Prove that if P ∈L(V) satisfies P = P, then trace P is a nonneg-
ative integer.

10. Prove that if V is an inner-product space and T ∈L(V), then

trace T ∗ = trace T.

11. Suppose V is an inner-product space. Prove that if T ∈L(V) is
a positive operator and trace T = 0, then T = 0.