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MTE-02
Assignment Booklet
Bachelor’s Degree Programme
Linear Algebra
School of Sciences
Indira Gandhi National Open University
New Delhi
2007
Dear Student,
Please read the section on assignments in the Programme Guide for Elective Courses that we sent you
after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous
evaluation, which would consist of two tutor-marked assignments, which are in this booklet.
Instructions for Formating Your Assignments
Before attempting the assignment please read the following instructions carefully.
1) On top of the first page of your answer sheet, please write the details exactly in the following format :
ROLL NO:………………………….
NAME:………………………….
ADDRESS:………………………….
…………………………..
…………………………..
…………………………..
COURSE CODE:………………………….
COURSE TITLE:………………………….
ASSIGNMENT NO:………………………
STUDY CENTRE:………………………...
DATE:…………………….………………
PLEASE FOLLOW THE FORMAT ABOVE STRICTLY TO FACILITATE EVALUATION
AND TO AVOID DELAY.
2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.
3) Leave a 4 cm. margin on the left, top and bottom of your answer sheet.
4) Your answers should be precise.
5) While solving problems, clearly indicate which part of which question is being solved.
6) The assignment responses are to be submitted to your Study Centre Coordinator by November, 2007,
not later.
Please keep a copy of your answer sheets.
Wish you luck!
2
ASSIGNMENT 1
(To be done after studying Blocks 1 and 2.)
Course Code : MTE-02
Assignment Code : MTE-02/TMA-1/2007
Total Marks : 100
1. Which of the following statements are true? Give reasons for your answer.
i)
The product of two upper triangular n × n matrices is upper triangular.
ii)
{1,
iii)
If k is a field, so is k × k.
iv)
The scalar product of two vectors is the magnitude of their vector product.
v)
If (x1, y1) and (x2, y2) are linearly dependent elements of R2, then x1 y2 = x2 y1.
vi)
Q is a real vector space.
vii)
{R + a, R + b} is a basis for C R , where a, b  C.
- 1 , ¥ , Qazi} is a set.
viii) If T : V  W is a one-one linear transformation, then V and W are isomorphic.
ix)
If S and T are two linear transformations for which S T is defined, then nullity ( S T )
= nullity (S).
x)
The elementary matrices are of 6 different types.
(20)
2. a)
Consider S = {f : ] a, b [  R │f has at most finitely many discontinuities}, for some a, b  R.
Check whether S is a real vector space with respect to pointwise addition and scalar
multiplication of functions.
(6)
b)
Let S be a set and K a field. Define KS to be the set of all mappings from S to K. Let T  S.
Define K ST = {f  KS │f (t) = 0 " t  T}. Show that K ST is a subspace of KS.
Also check whether A = {f  Ks│f (s) = 0 except for finitely many s  S} is a subspace of KS.
(4)
c)
Let V = Fn, where F is a field, and W = {(x1, …, xn)  V │
n
å
i =1
x i = 0}. Obtain dimF V W .
Hence show that there is no subspace U of V such that W  U  V.
(4)
d)
Let V, V1, V2, W be vector spaces such that V = V1 + V2 and V2 = (V1 Ç V2) Å W. Prove that
V = V1 Å W.
(6)
3. a)
Obtain the conditions on l  C under which {(1,1,1), (1, l , l 2 ), (1, – l , l 2 ) is a basis of C3.
(4)
4
Find a basis of R
, where V is the subspace spanned by
V
{(1,3,2,5), (2,4,3,1), (5,11,8,7)}.
(6)
b)
4. a)
Determine which of the following functions are linear:
i)
T : R3  R2 : T (x, y, z) = (xy, yz)
ii)
T : R3  R2 : T (x, y, z) = (x + y, z)
3
éx
T : R3  M3 (R) : T (x, y, z) = ê0
ê
ëê0
iii)
y
x
0
zù
yú
ú
x ûú
For those T that are linear, obtain R(T), Ker T, rank and nullity of T.
b)
5. a)
(10)
Show that if X and Y are subspaces of a vector space V such that V X and V Y are finitedimensional, then V
is also finite-dimensional.
(5)
(X Ç Y)
Consider the linear transformation T : R2  R3 : T(x, y) = (2x – y, x + y, x – 3y). Obtain
[T]B , B , where B1 = {(2, 1), (0, 2)} and B2 = {(1, 0, 2), (0, 1, 1), (1, 0, 3)}.
1
2
Also obtain the matrix of change of basis to the basis B1 = {(1, –1), (1, 1)} of R2. Hence obtain
(7)
[T]B1 ,B2 .
b)
c)
6. a)
é1 2 3ù
Using Theorem 7, Unit 7, check whether A = ê1 0 2ú is invertible. If it is, find its inverse.
ê
ú
êë0 1 0 úû
Otherwise change a11 to make A invertible.
(5)
Give an example of a linear transformation from R4 to R4 whose minimum polynomial is
(x2 – 1) (x + 1). What is its characteristic polynomial?
(3)
Let V and W be vector spaces over F, with bases {v1, …, vn}and {w1, …., wm}, respectively.
Let A be the matrix of a linear transformation T : V  W with respect to these bases. Prove
that v  Ker T iff v =
n
x
i
v i , where x = [x1, …, xn]t satisfies Ax = 0.
(6)
i 1
b)
7. a)
b)
Find a basis for the null space of the linear system given below:
x1 + 2x2 + x3 + 2x4 = 0, 3x1 + 6x2 + 4x3 = x4, –2x1 –4x2 + 5x3 = 53x4
(4)
Find the vector equation of the plane determined by (2, 0, –1), (–1, 1, 2) and (2, 2, –3). Also
find the point at which this plane intersects the plane r = a i + (1 + a) j + (a – 2) k.
(5)
For a = (–1, 2, 0) and b = (3, –2, 5), find the angle between a + b and a – b.
4
(5)
ASSIGNMENT 2
(To be done after studying Blocks 3 and 4.)
Course Code : MTE-02
Assignment Code : MTE-02/TMA-2/2007
Total Marks : 100
1. Which of the following statements are true? Give reasons for your answer.
i)
For a square matrix A, rank (A) = rank (Adj (Adj(A))).
ii)
The minimum polynomial of a linear operator whose characteristic polynomial is
(x – )m (x – )n must be (x – ) (x – ), where ,   R.
iii)
The set of eigenvalues of a linear operator is a subspace of R.
iv)
If f : V  R is a linear functional, where V is a real vector space, then
φ: V × V  R : φ (v, w) = f (v) – f (w) is an inner product.
v)
There is no linear operator which is unitary as well as self-adjoint.
vi)
A quadratic equation in one variable represents a conic.
2. a)
b)
(12)
é2 5 0 ù
Obtain Adj(A), where A = ê2 2 - 1ú . If Adj(A) is invertible, obtain its inverse as well as
ê
ú
êë4 7 0 úû
A–1. If Adj(A) is not invertible, obtain the eigenvalues of Adj(A).
(6)
Solve the following system of equations by Gaussian elimination:
2y + 2z – t = 6, x + 2y + 5z = 12, x + 4y + 7z = 2t + 5
At each stage write down the operation being applied.
c)
Explain why the following linear system can be solved by Cramer’s Rule, and use the rule for
solving it.
x + 2y – z + 5 = 0, 3x – y + 15z = –1 , y –5x = 9
3. a)
b)
(4)
(5)
Give examples of matrices A and B such that their characteristic polynomials are the same but
their minimum polynomials are different.
(4)
é- 2 0 6 ù
Determine whether A = ê- 1 1 2ú is diagonalisable. If it is, find matrices D and P such that
ê
ú
êë- 2 0 5 úû
D = P–1 AP, and find A40. If A is not diagonalisable, obtain Adj (A).
[Hint: Remember that an n × n matrix A is diagonalisable if the sum of the dimensions of its
eigenspaces is n.]
(12)
5
c)
The companion matrix of the polynomial p(x) = a0 + a1x +
+ an – 1xn – 1 + xn is the n × n matrix
0 - a0 ù
é0 0
ê1 0
0 - a1 ú
ê
ú
ê0 1
ú
0
ê
ú
A=ê
ú
ê
ú
ê
ú
0 - a n- 2 ú
ê0 0
êë0 0
1 - a n- 1 ú
û
Show that the characteristic equation of A is p(x) = 0.
[Hint: By applying appropriate transformations, show that
0
0
0
p( l )
-1
0
0
*
0
-1
│λ I – A │ =
, where * denotes some entries.]
0
0
*
0
-1
*
(5)
d)
For A  Mn (R), define TA : Mm×n (R)  Mm×n (R) : TA (B) = AB. Show that
i)
ii)
4. a)
c)
5. a)
(4)
Which of the following define an inner product on C2? Give reasons for your answers.
i)
ii)
b)
TA is a linear operator.
λ is an eigenvalue of TA iff λ is an eigenvalue of A.
<(x1, x2), (y1, y2)> = x1 y1 + x2 y2
<(x1, x2), (y1, y2)> = 2x1 y1 + i (x2 y1 – x1 y 2 ) + 2 x2 y 2
(2)
Use the Gram-Schmidt process to obtain an orthonormal basis for R3 from the basis
B = {(1,0,2), (0,1,1), (1,0,3)} with respect to the standard inner product.
(5)
For any subspace W of R4, with respect to the standard inner product, define
W ^ = {x  R4 │<x, w> = 0 " w  W}. Obtain W ^ , if
W = {(x1, x2, x3, x4)  R4 │ 2x1 + 3x2 + 5x3 + x4 = 0, x1 + x2 + x3 = 0}
(3)
Consider R3 [x], with the inner product <p(x), q(x)> =
1
ò p(x) q(x) dx.
-1
Define f: R3 [x]  R: f (p(x)) = p(0). Show that f is a linear functional, and find the unique
q(x)  R3 [x] such that f = < , q>.
(5)
b)
Find the adjoint of the differential operator D for the inner product space given in Q5 (a).
c)
Find a unitary matrix whose first two columns are
t
(8)
t
é 1 -1 1 ù é 1 1 ù
,
,
ê ,
ú , ê0,
ú .
2 2û
ë 3 3 3û ë
(2)
6
6. a)
Consider the quadratic form Q = 3x2 + 2y2+ 3z2 – 2xy – 2yz.
i)
ii)
iii)
iv)
v)
b)
7. a)
b)
Find the symmetric matrix associated with Q.
Find the rank of Q.
Obtain the orthogonal canonical reduction of Q.
Obtain the normal canonical reduction of Q.
Find the signature of Q.
(8)
Reduce x2 + 4xy – 2y2 = c to standard form, where c  R. Hence identify the conic it
represents.
(7)
Let A, B  Mn (R) such that det (I – AB) ≠ 0. Show that (I – BA)–1 exists.
(5)
Show that the set of n × n skew-symmetric matrices is a subspace of Mn (R).
(3)
7