MATH40003 Linear Algebra and Groups (2021)
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MATH40003 Linear Algebra and Groups (2021)
1. (a) For which values of α e R does the system of equations
(
2x - y - 3z = 3
αx - 4y + 2z = 5α for x, y, z e R
ì
ì
ì
-3x + 6y + 2z = -5 ìììì
have:
(i) no solutions?
(ii) exactly one solution?
(iii) infinitely many solutions?
In parts (ii) and (iii) you should give the solutions. (10 marks)
(b) Suppose V is a vector space over a field F . Suppose 0卜 W and W c V . Is W a subspace
of V? Give full details of the argument, assuming only axioms of a vector space/field.
(2 marks)
(c) Let S be the set of solutions to the system in (a) (where α e R is a constant). Which of the following sets are subspaces of R3 ? Does the answer depend on the value of α? Justify your answer fully.
(i) S .
(ii)
╱ 、 ╱ 、
| | | |
| | | |
| | | |
| | | |
| -2 | | -2 |
(iii)
╱ - 13 、 ╱ - 13 、
| 9 | | 9 |
| | | |
| 9 | | 9 |
| | | |
| 0 | | 0 |
(8 marks) (Total: 20 marks)
2. Let V be the R-vector space R4 . Let
B = , , , , and C = , .
(a) Show that W is a subspace of V .
┌╱ 、┐
' | | '
' | | '
(b) Show that B is a basis for V and find α2 where α 1 , α2 , α3 , α4 e R.
'(') α3 '(')
' | | '
' | | '
'| α4 |'去
(c) Let T : V → W be the linear map such that:
(3 marks)
(5 marks)
╱x 、 ╱ 2z 、
(i) Find Im(T) and dim(Im(T)).
(ii) Find 4 [T]去 . (5 marks)
(d) Let T, V, W be as above. Let V* be the R-vector space of linear transformations from V to
V . And let U* be the R-vector space of linear transformations from V to W . Let T* : V* → U* be such that T* (f) = T o f.
Show that
(i) T* is a linear transformation.
(ii) If B* is a basis for V* and C* is a basis for U* determine the rank of 4* [T*]去 * .
Any results from lectures used must be clearly stated. (7 marks)
(Total: 20 marks)
3. Let V be an R-vector space and let b1 , ..., bA be a basis for V with n > 2. We say an idempotent is a linear transformation T : V → V such that T2 = T o T = T .
(a) Denoting a general element of V by v = α 1 b1 + . . . + αA bA (for α 1 , . . . , αA e R), which of
the following maps are:
* Linear transformations?
* Idempotents?
Give clear justification in all cases.
i. S0 : V → V , where S0 (v) = 0卜 .
ii. S1 : V → V , where S1 (v) = b1 .
iii. S2 : V → V , where S2 (v) = α 1 b1 + α2 b2 .
iv. S3 = (S2 )2 = S2 o S2 .
v. S4 : V → V , where S4 (v) = α2 b1 + α1 b2 .
(11 marks) (b) Show that if T : V → V and S : V → V are linear transformations, then S o T is a linear
transformation. (2 marks)
(c) Suppose T : V → V and S : V → V are idempotent. Is S o T idempotent? Justify your answer fully giving a proof or counterexample. (4 marks)
(d) When can you construct an idempotent linear transformation T : V → V such that Im(T) = ker(T). Justify your answer fully, any results used from lectures must be clearly stated.
(3 marks) (Total: 20 marks)
4. (a) The sequences of real numbers (aA )A>1 and (bA )A>1 satisfy the relations: aA+1 = 3aA + 2bA and bA+1 = -4aA - 3bA for all n > 1.
(i) Write down a matrix A with the property that
A ╱b(a)A(A)\ = \
and compute its eigenvalues and eigenvectors (over R), showing the details of your computations.
(6 marks) (ii) Hence, or otherwise, give general expressions for aA and bA in terms of n, a1 and b1 .
(3 marks) (iii) Which values for a1 and b1 imply that both a17 = 3 and b18 = 12? (1 mark)
(b) (i) Suppose v1 , . . . , vJ e RA is an orthogonal set of non-zero vectors. Prove that for all
α 1 , . . . , αJ e R we have
J J
|↓ α;v; | = (↓ α;(2)|v; |2 ) .
;=1 ;=1
(2 marks)
(ii) Suppose A e MA (R) is symmetric. Let α be the maximum value of |λ| for all eigenvalues
λ of A.
● Using (b)(i), show that |Av| < α|v| for all v e RA and there is some non-zero v for which we have equality. Clearly state any results from the notes which you require.
● Deduce that sup{|Av| : |v| = 1} = α .
● State and prove a similar result for inf{|Av| : |v| = 1}.
(8 marks) (Total: 20 marks)
5. (a) Let n > 2 and let F be a field. Suppose A e MA (F) has exactly n non-zero entries x1 , . . . , xA , one in each row and one in each column of A.
(i) Prove that det(A) is equal to x1 x2 . . . xA or -x1 x2 . . . xA .
(ii) One of the zero entries in A is changed to 1, resulting in the matrix B. Express det(B )
in terms of det(A), explaining your answer.
(6 marks)
(b) Suppose C e M4 (F) and λ e F is an eigenvalue of C such that the eigenspace E乂 has
dimension 2. Prove that χ4 (x) = (x - λ)2g(x) for some polynomial g(x) with coefficients in F . (You may quote results from the notes, but you should not quote general results which immediately imply what is to be proved.) (6 marks)
(c) Let G be the symmetric group S6 and H = {h e G : {h(1), h(2)} = {1, 2}}.
(i) Show that H is a subgroup of G and determine |H|, explaining your answer.
(3 marks)
(ii) Prove that for g, k e G we have
gH = kH 令 {g(1), g(2)} = {k(1), k(2)}.
(iii) Using (c)(i) and (ii), compute the index of H in G in two different ways. (Throughout the question, results from the notes may be used if clearly stated.)
(3 marks) (2 marks)
(Total: 20 marks)
6. (a) Suppose (G, .) is a group and let H = {g2 : g e G}. Three of the following four statements are true and one is not true in general. Prove the ones which are true and give, with explanation, a counterexample for the other one.
(i) If G is abelian, then H is a subgroup of G. (ii) If G is finite and |G| is odd, then H = G.
(iii) If n > 4 and G is the symmetric group SA , then H = G.
(iv) If G is the dihedral group D2A , then H is a subgroup of G.
(8 marks)
Consider the following elements of S9 :
╱ 、 ╱ 、
| 3 4 5 6 7 8 9 2 1 | | 9 8 7 6 5 4 3 2 1 | .
(i) Write f, g, f_1 , g3 fgf2 in disjoint cycle form and state the orders of these. (4 marks) (ii) What are the cycle shapes of elements of order 3 in S9 ? How many elements of order 3
are there in S9 ? (You may leave your answers expressed in terms of binomial coefficients.) (4 marks)
(c) Suppose G is a finite group. Prove that the number of elements of order 5 in G is divisible by 4. State and prove a more general result about the number of elements of order p in G, where p is a prime number. (4 marks)
(Throughout the question, results from the notes may be used if clearly stated.)
(Total: 20 marks)
2022-05-07