1. True or false: If V is a vector space, then there is a unique vector x satisfying the
equation x + v = v for all vectors v in V .

A. True
B. False

2. True or false: If U =  ab  : a2 = b2, then U is a subspace of R2.

A. True
B. False

3. True or false: If A is an n × n matrix with real entries then A + AT is diagonalizable.

A. True
B. False

4. True or false: Every complex matrix is similar to an upper triangular matrix.

A. True
B. False

5. True or false: The matrices  20 2 i 1i  and  20 2 i 0i  are similar matrices.

A. True
B. False

6. Give an example of a possible candidate for the Jordan canonical form of a matrix
with distinct eigenvalues 4i and 2 with algebraic multiplicities 4 and 2 and geometric
multiplicities 2 and 1 respectively.

11. Suppose that A is a 7 × 7 matrix with characteristic polynomial p(λ) = (λ − 1)3(λ − 2)4,
that

dim(null((A − I)1)) = 1
dim(null((A − I)2)) = 2
and that
dim(null((A − 2I)1)) = 2
dim(null((A − 2I)2)) = 4:

Find the Jordan form of A.

12. Suppose that A is an n × n matrix whose only eigenvalue is λ. Show that there exists
an integer k such that (A − λI)k = 0.
(Hint: use Schur’s lemma).

13. Suppose that A is a 3 × 3 matrix, and let T : R3 ! R3 be the linear transformation
defined by T(x) = Ax. Suppose also that

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