MAT2611 Linear Algebra II Exam Questions And Answers
Section A
This section is for students in the current session and have prepared according to
... [Show More] tpresent text book Linear Algebra Done Right by Sheldon Axler.
Question 1 (a) Let
f = (x, x2
) : x ∈ R
g = (x,
√
x) : x ∈ R ∪ (x, −
√
x) : x ∈ R
h = (x, cos x + i sin x) : x ∈ R .
Which of the sets f , g and h is a function? If it is a function, indicate its domain, codomain,
range, whether it is injective and whether it surjective.List also the compositions, if possible,
and indicate whether the composite is injective or surjective.
(b) Show that a function A
f
−→ B is bijective if and only if there exists a unique function B
g
−→ A
such that f ◦ g = 1B and g◦ f = 1 A .
[Total Marks for Question 1: 4 + 6 = 10 marks]
Question 2 (a) Find the fifth roots of 1 + i.
(b) Find the values of λ ∈ C such that
λ(3 − 4i, −25) = (3 + 4i, 7 − 24i).
(c) Let m be a non-negative integer and p P ∈P m (C).
Show that p P ∈P m (R) if and only if there exist at least m + 1 real numbers x with p(x) ∈ R.
[Total Marks for Question 2: 5 + 3 + 12 = 20 marks]
Question 3 (a) Show that in the definition of a vector space V the condition about existence
of additive inverse can be replaced with the condition:
0v = 0, for all v V. ∈
Here the 0 on the left side of the equation is the scalar 0, and on right side of the equation
is the additive identity of V .
(b) Show that if U 1, U2, . . . , Um are finite dimensional subspaces of a vector space V such that:
Ui ∩ Uj =
(
{0}, if i 6= j,
Ui
, if i = j.
[TURN OVER]
3
then:
dim U1 ⊕ U2 ⊕ · · · U ⊕ m = dim U1 + dim U2 + · · · + dim Um .
Guess and prove a formula for dim(U1 + U2 + U3), in general when Ui ∩ Uj 6= {0}, for i 6= j.
State your reasons for the guess.
[Total Marks for Question 3: 10 + (20 + 10) = 40 marks]
Question 4
Show that for a vector space V , dim V < 2 if and only if for every f, g L ∈ (V ), g◦ f = f ◦ g.
[Total Marks for Question 4: 10 marks]
Question 5 (a) Suppose V is a finite dimensional vector space and T L ∈ (V ).
Show that dim range T ≥ m − 1, where T has m distinct eigenvalues.
(b) Given any T L ∈ (V ) on an inner product space V define:
[u, v] =
D
T (u), T (v)
E
. (†)
Does the function (u, v) 7→ [u, v] in (†) define an inner product?
If it does, prove it; if it does not provide an example.
Provide a necessary and sufficient condition which would make the function an inner product
on V .
(c) Find p P ∈ P3 (R) such that p(0) = 0 = p 0
(0) and
Z 1
0
|2 + 3x − p(x)|
2dx
is as small as possible.
[Total Marks for Question 5: 5 + 5 + 10 = 20 marks] [Show Less]