Download CM222, Linear Algebra Mock Test 3 Solutions 1. Let P2 denote the

CM222, Linear Algebra
Mock Test 3 Solutions
1. Let P2 denote the vector space of real polynomials of degree at most two,
and let L : P2 → P2 be the linear map defined by Lf = f 0 . Which of the
following are eigenvalues of L?
(a) −1
(b) 0
(c) 1
(d) 2
Solution: b)
Explanation: The matrix of the
{1, x, x2 } of P2 is

0
 0
0
linear map with respect to the basis

1 0
0 2 .
0 0
The characteristic polynomial is therefore x3 , whose only root is 0.
2. Let A be a complex n × n-matrix and let λ be an eigenvalue of A. Which
of the following are correct descriptions of the geometric multiplicity of λ?
(a) the nullity of A − λI
(b) the largest m such that (x − λ)m divides mA (x) (the minimal polynomial of A
(c) the largest possible number of eigenvectors for A with eigenvalue λ
in a linearly independent set
(d) the largest m such that (x − λ)m divides pA (x) (the characteristic
polynomial of A
Solution: a), c)
Explanation: a) is the definition.
c) is the same as a) since the eigenvalues with eigenvalue λ are precisely
the non-zero vectors in the null space of A − λI.
b) is incorrect, since for example the geometric multiplicity of the eigenvalue 1 for the 2 × 2 identity matrix is 2, but the minimial polynomial is
x − 1.
d) is the definition of the algebraic multiplicity.
3. Let A be a complex n × n-matrix. Which of the following are equivalent
to A being diagonalizable?
(a) A has n distinct eigenvalues.
(b) A satisfies its characteristic equation.
(c) A is similar to a diagonal matrix.
(d) Cn has a basis consisting of eigenvectors for A.
Solution: c), d)
Explanation: c) is the definition and we showed d) is equivalent.
a) is sufficient but not necessary for A to be diagonalizable.
b) holds for all square matrices.
4. Let A be a 4 × 4 complex matrix with minimal polynomial x3 + x2 . Which
of the following could be the characteristic polynomial of A?
(a) x4 + x2
(b) x4 + 2x3 + x2
(c) x4 + x3
(d) x3 + x2
Solution: b), c)
Explanation: The characteristic polynomial has degree 4, and it has the
same roots as the minimal polynomial, namely 0 and 1. Furthermore the
characteristic polynomial must be divisible by the minimal polynomial,
which is x2 (x + 1). The two remaining choices are possible since they have
the form x3 (x + 1) and x2 (x + 1)2 .
5. Let v1 = √12 (1, 0, i), v2 = √13 (i, 1, 1) and v3 = √16 (i, −2, 1). Given that
S = {v1 , v2 , v3 } is an orthonormal basis for C3 , determine which of the
following appear in the coordinate vector for w = (0, 1, 1) with respect to
S.
(a)
(b)
√2
3
√1
2
(c) −1
(d) i
Solution: a)
Explanation: The coordinates
are
√
√ given by hw, vi i for i = 1, 2, 3, and
are therefore −i/2, 2/ 3 and −1/ 6.