Extra Problems
To prepare for the exam, please first review my lecture notes, old homework,
old quizzes. DON’T do the exercises on this sheet before reviewing the basics.
The problems on this sheet are hard, but provide you a deeper understanding
of the concepts.
For numerical exercises, please find them in the textbook. The following are more conceptual ones.
1. Show that if AB is invertible, so is A.
2. Show that if AB is invertible, so is B.
3. Show that if A is an invertible upper triangular matrix, then A−1 is also upper triangular.
4. Let A be an n × n matrix. Show that if
2A4 − 5A2 + 4A + 2I = 0,
then A is invertible. Find A−1 .
5. Let A, B be matrices of sizes s × n, n × m, respectively. Show that
rankA + rankB ≤ n.
6. Let A, B be two n × n matrices. Suppose that
A + B = AB.
Show that I − A, I − B are invertible and AB = BA.
7. Let A be an n × n matrix.
(a) Show that
rankA ≥ rankA2 ≥ · · · ≥ rankAk ≥ rankAk+1 ≥ · · · .
(b) Show that the above inequality terminates at some integer l, i.e., there exists an
integer l such that
rankAl = rankAl+1 .
8. Let V be a vector space. Show that −(−v) = v for all v ∈ V .
1
9. Let ∞ and −∞ denote two distinct objects, neither of which is in R. Define an addition
and scalar multiplication on R ∪ {∞} ∪ {−∞} as follows:
The sum and product of two real numbers is as usual. For t a scalar,


−∞if t < 0
t∞ = 0
if t = 0


∞
if t > 0


if t < 0
∞
t(−∞) = 0
if t = 0


−∞ if t > 0
t+∞=∞+t=∞
t + (−∞) = (−∞) + t = −∞
∞+∞=∞
(−∞) + (−∞) = −∞
∞ + (−∞) = 0.
Is R ∪ {∞} ∪ {−∞} a vector space under the operations defined above? Explain.
10. Let V be a vector space of dimension n. Show that any set of n + 1 vectors in V is
linearly dependent.
11. Let U, W be two subspaces of V and U ⊂ W . Show that dim U ≤ dim W .
12. Let R[0,1] denote the vector space of all real-valued functions defined on the interval
[0, 1]. Suppose b ∈ R. Show that the set of all continuous real-valued functions f on
the interval [0, 1] such that
Z
1
f dx = b
0
is a subspace of R[0,1] if and only if b = 0.
13. Let V be a vector space and U1 , U2 two subspaces of V . Show that U1 ∩U2 is a subspace
of V .
14. Let B = {b1 , b2 }, C = {c1 , c2 }, D = {d1 , d2 } be bases of a 2-dimensional vector space.
Write an equation that relates the matrices P , P and P . Justify your result.
C←B D←C
2
D←B