Mathematical Physics I: Assignment 3

HRI Graduate School

August - December 2009

13 October 2009
To be returned in the class on 22 October 2009

The deadline for the submission of the solutions of this assignment will be strictly enforced.
No marks will be given if the assignment is not returned in time.

You are free to discuss the solutions with friends, seniors and consult any books. However,
you should understand and be clear about every step in the answers. Marks may be
reduced if you have not understood what you have written even though the answer is
correct.

Let me know if you find anything to be unclear or if you think that something is wrong
in any of the questions.

   
2 0
1. Suppose A is an antisymmetric tensor, S is a symmetric tensor, C is an
  0   2
0 2
arbitrary tensor and D is an arbitrary tensor. Prove
2 0
(i) Aij Sij = 0
(ii) Aij Cij = Aij C[ij]
(iii) Sij Dij = Sij D(ij)
where we have defined
1 1
C[ij] (Cij Cji ); D(ij) (Dij + Dji )
2 2
Also prove
(iv) Cij = C(ij) + C[ij] and Dij = D(ij) + D[ij]

[2 + 2 + 2 + 1 = 7]

2. A basis {eqi } is called a coordinate basis if there exists curvilinear coordinates q i such
that
xj
eq i = ej
q i
where xj are the Cartesian coordinates and {ej } are the corresponding orthonormal basis.
Consider the polar coordinates (r, ) in two dimensions.
(i) Calculate the coordinate basis {er , e } (using the above equation) in terms of the
Cartesian basis {i, j}.
(ii) Calculate the orthonormal basis {er , e } in terms of {i, j}.
(iii) Assume that the basis {er , e } is a coordinate basis, i.e., there exists a coordinate
system (, ) such that
x y x y
er e = i+ j; e e = i+ j

1
 Then compute the quantities

, , ,
x y x y
Show that
2 2
6=
yx xy
What do you conclude from this?

[2 + 2 + 5 = 9]

3. Derive the form of the Laplacian of a scalar field in a general coordinate system. What
is its form in orthogonal coordinate systems?

[4]

4. Show that, in a general coordinate system, curl of a gradient of a scalar field is identically
zero.

[2]
ij
5. Show that gij;k = g;k = 0.

[4]

6. Are the following coordinate transformations (x, y) (, ) good ones? Compute the
Jacobian and list any points at which the transformations fail.
(i) = x, = 1
(ii) = (x2 + y 2 )1/2 , = tan1 (y/x)
(iii) = ln x, = y
(iv) = tan1 (y/x), = (x2 + y 2 )1/2 .

[2 4 = 8]

7. Show that if U i V;ik = W k , then U i Vk;i = Wk .

[2]

8. (i) Calculate all the Christoffel symbols of second kind for the metric

ds2 = dt2 a2 (t)dx2

where a(t) is an arbitrary function of t.

(ii) Write the explicit form of the gradient of a scalar field (F )i = g ik F/dxk , the
divergence of a vector field V = V;ii and the Laplacian F in this coordinate
system.
(iii) If a = 1, the above metric reduces to a simple Minkowskian one. Write the gradient,
divergence and Laplacian for the Minkowski metric.

[6 + 6 + 2 = 14]

2
 Mathematical Physics I: Assignment 2
HRI Graduate School
August - December 2009

3 September 2009
To be returned in the class on 14 September 2009

The deadline for the submission of the solutions of this assignment will be strictly enforced.
No marks will be given if the assignment is not returned in time.

You are free to discuss the solutions with friends, seniors and consult any books. However,
you should understand and be clear about every step in the answers. Marks may be
reduced if you have not understood what you have written even though the answer is
correct.

Let me know if you find anything to be unclear or if you think that something is wrong
in any of the questions.

1. Suppose {1 , . . . , n } spans a vector space V . Prove:

(i) If V , then {, 1 , . . . , n } is linearly dependent and spans V .
(ii) If i is a linear combination of vectors (1 , 2 , . . . , i1 ), then {1 , . . . , i1 , i+1 , . . . , n }
spans V .

[2 + 2 = 4]

  
2 2 x 3x + 2y
2. Let f : R R be given by f =
y 6x 4y
(i) Find a basis for the kernel of f .
(ii) Find a basis for the image of f .
(iii) Find the matrix for f with respect to the usual basis for R2 .

[2 + 2 + 2 = 6]

3. (i) Let V be a vector space and W V a subspace. Show that the relation

v v0 W

defines an equivalence relation on V .

(ii) Visualize
 geometrically
 the partition into equivalence classes in the case V = R2 and
1
W = span .
1

[3 + 2 = 5]

4. Let V be a vector space and U1 , U2 be two proper subspaces of V . Then show that U1 U2
is a subspace of V if and only if either U1 U2 or U2 U 1.

[3]

1
5. Let f : R3 R3 denote rotation about the line x = y = z by /2. Find a basis B for
R3 for which it is easy to express f in terms of a matrix, and find the matrix for f with
respect to your basis B.

[4]

6. Find a basis for the vector space defined by {v = (x, y, z, w) R4 |3x y z + w = 0}

and determine its dimension.

[2]

7. In each of the following parts, determine whether or not the set V with the given opera-
tions of addition and scalar multiplication, is a vector space over the given field F .
(i) V = {(a1 , a2 )|a1 , a2 R}, F = R, with vector sum (a1 , a2 ) + (b1 , b2 ) = (a1 + b1 , a2 + b2 )
and scalar multiplication c(a1 , a2 ) = (ca1 + c 1, ca2 + c 1), c R.
(ii) V = P (C) (the space of polynomial functions over C), F = C, with vector sum
(f + g)(z) = f (z) + g(z), and scalar multiplication (cf )(z) = f (cz), c C.
(iii) Let V be the set of functions f : Z R such that f (n) > 0 for every integer n, with
vector sum (f + g)(n) = f (n)g(n) and scalar multiplication (cf )(n) = (f (n))c , c R.
(iv) Let V be the set of functions f : R R, F = R, with vector sum (f + g)(t) =
1
2
(f (t) + f (t) + g(t) + g(t)), and scalar multiplication (cf )(t) = cf (t), c R.
(v) Let V = {(z1 , z2 )|z1 , z2 C}, F = C, with vector sum (z1 , z2 ) + (z10 , z20 ) = (z1 +
z10 , z2 + z20 ) and scalar multiplication c(z1 , z2 ) = (cz1 , c z2 ), c C, where c is the complex
conjugate of c.

[4 4 = 16]

8. (i) Let v R3 be a nonzero vector. Show that v : R3 R3 defined by v (w) = v w

is a homomorphism of vector spaces over R, where v w is the vector (cross) product of
v and w. Determine the kernel and the image of v .
(ii) Let 0 < 2 be an angle. Let R : R2 R2 be the map that associates to a
vector v the vector R (v) obtained performing a (counterclockwise) rotation centered at
the origin of angle. Show that R is a homomorphism of vector spaces over R. Determine
kernel and image of R .

[3 + 3 = 6]

9. Determine whether the function f : V W is a linear transformation.

(i) Let V = W = P (F ), the space of polynomial over a field F . Define f ((x)) =
x(x2 + 1) + (x3 1)(1), V .
(ii) Let V = W = M 2,2 (C). Define f (A) = iAT 4A, A V . (Here AT is the transpose
of the matrix A.)
(iii) Let V be a finite-dimensional vector space over a field F . Let n = dim V and let
W = F n . Let {x1 , . . . , xn } be a basis of V . Define f (x) = (c1 , c2 c1 , c3 c2 , c4
c3 , . . . , cj cj1 , . . . , cn cn1 ) for x = c1 x1 + c2 x2 + . . . + cn xn V with (c1 , . . . , cn F ).
(iv) Let V = F 3 and W = F 2 , where F is a field. Let {x1 , x2 , x3 } be a basis of V .
Define f (c1 x1 + c2 x2 + c3 x3 ) = (c21 + c2 , c2 c3 ), c1 , c2 , c3 F.

[2 4 = 8]

2
10. Let f : P2 (C) M 2,2 (C) be the linear transformation defined by
 
3(i) i(0)
f ((x)) = , (x) P2 (C)
(i) 0 (0)

where P2 (C) is the set of polynomials over C with degree 2.

       
2 0 1 0 1 1 0 0 0
Let = {1, x 1, x ix} and = , , , .
1 0 1 0 0 0 0 1
Show that and form bases of P2 (C) and M 2,2 (C) respectively. Compute the matrix
representation of f with respect to the ordered bases and .

[4]

11. Let V = P3 (R) and W = M 2,2 (R), and let f L(V, W ) such that
 
3 2 d + c a
f (ax + bx + cx + d) = , a, b, c, d R
b c

Find f 1 (y) for any vector y W .

[5]

12. Let f L(P3 (R), P3 (R)) be defined by

f (ax3 + bx2 + cx + d) = ax3 + (a 3b + c + d)x2 + (2a b c 2d)x + d, a, b, c, d R.

(i) Find the characteristic polynomial and all eigenvalues of f .

(ii) Determine whether f is diagonalizable. Justify your answer.
(iii) Find a basis for the eigenspace E1 of f corresponding to the eigenvalue 1.

[3 + 3 + 2 = 8]

13. Let f, g L(V, V ), where V is a finite-dimensional vector space. Assume that f is

invertible.
(i) Prove that f is diagonalizable if and only if f 1 is diagonalizable.
(ii) Prove that g is diagonalizable if and only if f gf 1 = f g f 1 is diagonalizable.
(iii) Assume V to be a finite-dimensional vector space over C. If f is diagonalizable, prove
that there exists a diagonalizable h L(V, V ) such that h2 = f .

[2 + 3 + 2 = 7]

14. Let g and f be two linear operators on a finite-dimensional vector space V . If gji , fji are
matrix representations of g and f respectively, find the matrix representation of g f .

[2]

3
15. Let V = U W , and let = + be the unique decomposition of a vector V into
a sum of vectors from U and W Define the projection operators PU : V
U, PW : V W by
PU () = ; PW () =
Show that
(i) PU2 = PU and PV2 = PV .
(ii) Show that if P : V V is a linear operator satisfying P 2 = P , then there exists a
subspace U 0 such that P = PU 0 .
Hint: Set U 0 = {|P () = } and W 0 = {|P () = 0} and show that these are complementary subspaces
such that P = PU 0 and PW 0 = I P

[2 + 4 = 6]

16. Let f : R4 R4 be a linear operator whose effect on a basis {e1 , e2 , e3 , e4 } is

f (e1 ) = 2e1 e4
f (e1 ) = 2e1 + e4
f (e1 ) = 2e1 + e4
f (e1 ) = e1

Find a basis for Ker f and Im f and calculate the rank and nullity of f .

[4]

17. Let V be the vector space of 2 2 matrices with the usual basis
        
1 0 0 1 0 0 0 0
, , , ,
0 0 0 0 1 0 0 1
 
1 2
Let M = and g be the linear operator on V defined by g(A) = MA. Find the
3 4
matrix representation of g relative to the above usual basis of V .

[2]

18. Let P4 (C) be the vector space of polynomials having degree 4 over a variable t. Let
D : P4 (C) P4 (C) and X : P3 (C) P4 (C) be the derivative and multiplication-by-t
operators, respectively. Choose {1, t, t2 , t3 } as the basis of P3 (C) and {1, t, t2 , t3 , t4 } as
the basis of P4 (C).
(i) Find the matrix representations of D and X.
(ii) Use the matrix of D so obtained to find the first, second, third, fourth and fifth
derivatives of a general polynomial of degree 4.

[3 + 5 = 8]

4
 Mathematical Physics I: Assignment 2
HRI Graduate School
August - December 2009

3 September 2009
To be returned in the class on 14 September 2009

The deadline for the submission of the solutions of this assignment will be strictly enforced.
No marks will be given if the assignment is not returned in time.

You are free to discuss the solutions with friends, seniors and consult any books. However,
you should understand and be clear about every step in the answers. Marks may be
reduced if you have not understood what you have written even though the answer is
correct.

Let me know if you find anything to be unclear or if you think that something is wrong
in any of the questions.

1. Suppose {1 , . . . , n } spans a vector space V . Prove:

(i) If V , then {, 1 , . . . , n } is linearly dependent and spans V .
(ii) If i is a linear combination of vectors (1 , 2 , . . . , i1 ), then {1 , . . . , i1 , i+1 , . . . , n }
spans V .

[2 + 2 = 4]

  
2 2 x 3x + 2y
2. Let f : R R be given by f =
y 6x 4y
(i) Find a basis for the kernel of f .
(ii) Find a basis for the image of f .
(iii) Find the matrix for f with respect to the usual basis for R2 .

[2 + 2 + 2 = 6]

3. (i) Let V be a vector space and W V a subspace. Show that the relation

v v0 W

defines an equivalence relation on V .

(ii) Visualize
 geometrically
 the partition into equivalence classes in the case V = R2 and
1
W = span .
1

[3 + 2 = 5]

4. Let V be a vector space and U1 , U2 be two proper subspaces of V . Then show that U1 U2
is a subspace of V if and only if either U1 U2 or U2 U 1.

[3]

1
5. Let f : R3 R3 denote rotation about the line x = y = z by /2. Find a basis B for
R3 for which it is easy to express f in terms of a matrix, and find the matrix for f with
respect to your basis B.

[4]

6. Find a basis for the vector space defined by {v = (x, y, z, w) R4 |3x y z + w = 0}

and determine its dimension.

[2]

7. In each of the following parts, determine whether or not the set V with the given opera-
tions of addition and scalar multiplication, is a vector space over the given field F .
(i) V = {(a1 , a2 )|a1 , a2 R}, F = R, with vector sum (a1 , a2 ) + (b1 , b2 ) = (a1 + b1 , a2 + b2 )
and scalar multiplication c(a1 , a2 ) = (ca1 + c 1, ca2 + c 1), c R.
(ii) V = P (C) (the space of polynomial functions over C), F = C, with vector sum
(f + g)(z) = f (z) + g(z), and scalar multiplication (cf )(z) = f (cz), c C.
(iii) Let V be the set of functions f : Z R such that f (n) > 0 for every integer n, with
vector sum (f + g)(n) = f (n)g(n) and scalar multiplication (cf )(n) = (f (n))c , c R.
(iv) Let V be the set of functions f : R R, F = R, with vector sum (f + g)(t) =
1
2
(f (t) + f (t) + g(t) + g(t)), and scalar multiplication (cf )(t) = cf (t), c R.
(v) Let V = {(z1 , z2 )|z1 , z2 C}, F = C, with vector sum (z1 , z2 ) + (z10 , z20 ) = (z1 +
z10 , z2 + z20 ) and scalar multiplication c(z1 , z2 ) = (cz1 , c z2 ), c C, where c is the complex
conjugate of c.

[4 4 = 16]

8. (i) Let v R3 be a nonzero vector. Show that v : R3 R3 defined by v (w) = v w

is a homomorphism of vector spaces over R, where v w is the vector (cross) product of
v and w. Determine the kernel and the image of v .
(ii) Let 0 < 2 be an angle. Let R : R2 R2 be the map that associates to a
vector v the vector R (v) obtained performing a (counterclockwise) rotation centered at
the origin of angle. Show that R is a homomorphism of vector spaces over R. Determine
kernel and image of R .

[3 + 3 = 6]

9. Determine whether the function f : V W is a linear transformation.

(i) Let V = W = P (F ), the space of polynomial over a field F . Define f ((x)) =
x(x2 + 1) + (x3 1)(1), V .
(ii) Let V = W = M 2,2 (C). Define f (A) = iAT 4A, A V . (Here AT is the transpose
of the matrix A.)
(iii) Let V be a finite-dimensional vector space over a field F . Let n = dim V and let
W = F n . Let {x1 , . . . , xn } be a basis of V . Define f (x) = (c1 , c2 c1 , c3 c2 , c4
c3 , . . . , cj cj1 , . . . , cn cn1 ) for x = c1 x1 + c2 x2 + . . . + cn xn V with (c1 , . . . , cn F ).
(iv) Let V = F 3 and W = F 2 , where F is a field. Let {x1 , x2 , x3 } be a basis of V .
Define f (c1 x1 + c2 x2 + c3 x3 ) = (c21 + c2 , c2 c3 ), c1 , c2 , c3 F.

[2 4 = 8]

2
10. Let f : P2 (C) M 2,2 (C) be the linear transformation defined by
 
3(i) i(0)
f ((x)) = , (x) P2 (C)
(i) 0 (0)

where P2 (C) is the set of polynomials over C with degree 2.

       
2 0 1 0 1 1 0 0 0
Let = {1, x 1, x ix} and = , , , .
1 0 1 0 0 0 0 1
Show that and form bases of P2 (C) and M 2,2 (C) respectively. Compute the matrix
representation of f with respect to the ordered bases and .

[4]

11. Let V = P3 (R) and W = M 2,2 (R), and let f L(V, W ) such that
 
3 2 d + c a
f (ax + bx + cx + d) = , a, b, c, d R
b c

Find f 1 (y) for any vector y W .

[5]

12. Let f L(P3 (R), P3 (R)) be defined by

f (ax3 + bx2 + cx + d) = ax3 + (a 3b + c + d)x2 + (2a b c 2d)x + d, a, b, c, d R.

(i) Find the characteristic polynomial and all eigenvalues of f .

(ii) Determine whether f is diagonalizable. Justify your answer.
(iii) Find a basis for the eigenspace E1 of f corresponding to the eigenvalue 1.

[3 + 3 + 2 = 8]

13. Let f, g L(V, V ), where V is a finite-dimensional vector space. Assume that f is

invertible.
(i) Prove that f is diagonalizable if and only if f 1 is diagonalizable.
(ii) Prove that g is diagonalizable if and only if f gf 1 = f g f 1 is diagonalizable.
(iii) Assume V to be a finite-dimensional vector space over C. If f is diagonalizable, prove
that there exists a diagonalizable h L(V, V ) such that h2 = f .

[2 + 3 + 2 = 7]

14. Let g and f be two linear operators on a finite-dimensional vector space V . If gji , fji are
matrix representations of g and f respectively, find the matrix representation of g f .

[2]

3
15. Let V = U W , and let = + be the unique decomposition of a vector V into
a sum of vectors from U and W Define the projection operators PU : V
U, PW : V W by
PU () = ; PW () =
Show that
(i) PU2 = PU and PV2 = PV .
(ii) Show that if P : V V is a linear operator satisfying P 2 = P , then there exists a
subspace U 0 such that P = PU 0 .
Hint: Set U 0 = {|P () = } and W 0 = {|P () = 0} and show that these are complementary subspaces
such that P = PU 0 and PW 0 = I P

[2 + 4 = 6]

16. Let f : R4 R4 be a linear operator whose effect on a basis {e1 , e2 , e3 , e4 } is

f (e1 ) = 2e1 e4
f (e1 ) = 2e1 + e4
f (e1 ) = 2e1 + e4
f (e1 ) = e1

Find a basis for Ker f and Im f and calculate the rank and nullity of f .

[4]

17. Let V be the vector space of 2 2 matrices with the usual basis
        
1 0 0 1 0 0 0 0
, , , ,
0 0 0 0 1 0 0 1
 
1 2
Let M = and g be the linear operator on V defined by g(A) = MA. Find the
3 4
matrix representation of g relative to the above usual basis of V .

[2]

18. Let P4 (C) be the vector space of polynomials having degree 4 over a variable t. Let
D : P4 (C) P4 (C) and X : P3 (C) P4 (C) be the derivative and multiplication-by-t
operators, respectively. Choose {1, t, t2 , t3 } as the basis of P3 (C) and {1, t, t2 , t3 , t4 } as
the basis of P4 (C).
(i) Find the matrix representations of D and X.
(ii) Use the matrix of D so obtained to find the first, second, third, fourth and fifth
derivatives of a general polynomial of degree 4.

[3 + 5 = 8]

4
 Mathematical Physics I: Assignment 3
HRI Graduate School
August - December 2009

13 October 2009
To be returned in the class on 22 October 2009

The deadline for the submission of the solutions of this assignment will be strictly enforced.
No marks will be given if the assignment is not returned in time.

You are free to discuss the solutions with friends, seniors and consult any books. However,
you should understand and be clear about every step in the answers. Marks may be
reduced if you have not understood what you have written even though the answer is
correct.

Let me know if you find anything to be unclear or if you think that something is wrong
in any of the questions.

   
2 0
1. Suppose A is an antisymmetric tensor, S is a symmetric tensor, C is an
  0   2
0 2
arbitrary tensor and D is an arbitrary tensor. Prove
2 0
(i) Aij Sij = 0
(ii) Aij Cij = Aij C[ij]
(iii) Sij Dij = Sij D(ij)
where we have defined
1 1
C[ij] (Cij Cji ); D(ij) (Dij + Dji )
2 2
Also prove
(iv) Cij = C(ij) + C[ij] and Dij = D(ij) + D[ij]

[2 + 2 + 2 + 1 = 7]

2. A basis {eqi } is called a coordinate basis if there exists curvilinear coordinates q i such
that
xj
eq i = ej
q i
where xj are the Cartesian coordinates and {ej } are the corresponding orthonormal basis.
Consider the polar coordinates (r, ) in two dimensions.
(i) Calculate the coordinate basis {er , e } (using the above equation) in terms of the
Cartesian basis {i, j}.
(ii) Calculate the orthonormal basis {er , e } in terms of {i, j}.
(iii) Assume that the basis {er , e } is a coordinate basis, i.e., there exists a coordinate
system (, ) such that
x y x y
er e = i+ j; e e = i+ j

1
 Then compute the quantities

, , ,
x y x y
Show that
2 2
6=
yx xy
What do you conclude from this?

[2 + 2 + 5 = 9]

3. Derive the form of the Laplacian of a scalar field in a general coordinate system. What
is its form in orthogonal coordinate systems?

[4]

4. Show that, in a general coordinate system, curl of a gradient of a scalar field is identically
zero.

[2]
ij
5. Show that gij;k = g;k = 0.

[4]

6. Are the following coordinate transformations (x, y) (, ) good ones? Compute the
Jacobian and list any points at which the transformations fail.
(i) = x, = 1
(ii) = (x2 + y 2 )1/2 , = tan1 (y/x)
(iii) = ln x, = y
(iv) = tan1 (y/x), = (x2 + y 2 )1/2 .

[2 4 = 8]

7. Show that if U i V;ik = W k , then U i Vk;i = Wk .

[2]

8. (i) Calculate all the Christoffel symbols of second kind for the metric

ds2 = dt2 a2 (t)dx2

where a(t) is an arbitrary function of t.

(ii) Write the explicit form of the gradient of a scalar field (F )i = g ik F/dxk , the
divergence of a vector field V = V;ii and the Laplacian F in this coordinate
system.
(iii) If a = 1, the above metric reduces to a simple Minkowskian one. Write the gradient,
divergence and Laplacian for the Minkowski metric.

[6 + 6 + 2 = 14]

2
 Mathematical Physics I: Quiz 1
HRI Graduate School
August - December 2009

25 August 2009

1. Show that the infinite dimensional coordinate space R is isomorphic with a proper
subspace of itself.

[4]

2. On the vector space P (t) of polynomials with real coefficients over a variable t, let X
be the operation of multiplying the polynomial by t, and let D be the operation of
differentiation, i.e., X : P (t) P (t) such that

X() = t(t)

and D : P (t) P (t) such that

d(t)
D() = ,
dt
where P (t).
Show that (i) both of these are linear operators over P (t) and (ii) that DX XD = 1,
where 1 is the identity operator.

[3 + 3 = 6]

1
 Mathematical Physics I: Mid-term Examination
HRI Graduate School
August - December 2009

12 October 2009
Duration: 3 hours

The paper is of 100 marks. Attempt all the questions.

You are free to consult your class notes during the examination.

Let me know if you find anything to be unclear or if you think that something is wrong
in any of the questions.

1. Compute the product in the given ring.

(i) [12][16] in Z24
(ii) [16][3] in Z32
(iii) [11][-4] in Z15
(iv) [20][-8] in Z26
(v) ([2], [3]) ([3], [5]) in Z5 Z9
(vi) ([3], [5]) ([2], [4]) in Z4 Z11

[2.5 4 + 3.5 2 = 17]

2. Prove the following results for a group (G, ).

(i) The identity element e is unique.
(ii) Each a G has a unique inverse a1 .

[4 + 4 = 8]

3. Show that every element of SU (2) has the form

 
a b
c d

where a = d and b = c . Remember that SU (2) is the group of 2 2 unitary matrices

with determinant 1 and denotes complex conjugation.

[6]

4. The commutator [f, g] of two operators in L(V, V ) is defined as [f, g] = f g gh. Prove
the Jacobi identity [[f, g], h] + [[g, h], f ] + [[h, f ], g] = 0.

[6]

1
5. Let p1 and p2 be two linear operators on V such that p21 = p1 and p22 = p2 . Such
operators are termed projection. Show that p1 + p2 is a projection operator if and only
if p1 p2 = p2 p1 = 0.

[6]

6. Let f be a linear operator on V . One can define the exponential of a linear operator
through the convergent infinite series

X fj
ef exp(f ) =
j=0
j!

Suppose f : R2 R2 such that f (x, y) = (y, x). Show that

ef = f sin + 1 cos

where is a scalar.
What is the result of applying ef on (x, y)?

[10]

7. Let v R3 be a nonzero vector. Show that v : R3 R3 defined by v (w) = v w is

a homomorphism of vector spaces over R, where v w is the vector (cross) product of v
and w. Determine the kernel and the image of v .

[6]

8. Assume that f : V W is a linear transformation, where V = M 2,2 (R) is the space of

2 2 real matrices and W = P2 (R) is the space of real polynomials having degree 2.
Suppose that
       
0 1 2 1 0 1 1 2 0 0
f = t t, f = 3t, f = t + 4, f = t2 .
1 0 0 1 0 0 0 1
 
a b
Determine f for all real numbers a, b, c, d.
c d

[8]

9. Let V = P3 (R) and W = M 2,2 (R), and let f L(V, W ) such that
 
3 2 d + c a
f (ax + bx + cx + d) = , a, b, c, d R
b c

Find f 1 (y) for any vector y W .

[8]

2
10. Let V be the vector space of 2 2 matrices with the usual basis
        
1 0 0 1 0 0 0 0
, , , ,
0 0 0 0 1 0 0 1
 
1 2
Let M = and g be the linear operator on V defined by g(A) = MA. Find the
3 4
matrix representation of g relative to the above usual basis of V .

[6]

11. Let V = C4 with the standard inner product. Let

W = {x = (x1 , x2 , x3 , x4 ) V | 2x1 x3 = 0, x1 ix2 + x4 = 0}.

(i) Find an orthonormal basis for W .

(ii) Find an orthonormal basis for W .

[7 + 7 = 14]

12. A normalized wave function (x) = dx (x)(x) =

P R
n=0 an en (x) satisfies the condition
1. The en areRthe orthonormal basis functions in the infinite-dimensional Hilbert space
which satisfy dx ei (x)ej (x) = ij . The expansion coefficients an are known as prob-
ability amplitudes. We may define a density matrix with elements ij = ai aj . Show
that
(2 )ij = ij
or equivalently
2 =

[5]

3
 Mathematical Physics I: Quiz 2
HRI Graduate School
August - December 2009

3 September 2009

1. Assume that f : V W is a linear transformation, where V = M 2,2 (R) is the space of

2 2 real matrices and W = P2 (R) is the space of real polynomials having degree 2.
Suppose that
       
0 1 2 1 0 1 1 2 0 0
f = t t, f = 3t, f = t + 4, f = t2 .
1 0 0 1 0 0 0 1
 
a b
Determine f for all real numbers a, b, c, d.
c d

[4]

2. In the following, determine whether the vector spaces V and W are isomorphic. Justify
your answers.
(i) Let V = {A M 3,3 (C) | A = AT } and W = {A M 3,3 (C) | A = AT }. AT denotes
the transpose of A.
(ii) Let V = {f (t) P5 (C) | f (t) = f (t)} and W = P3 (C). Note that Pn (C) is the
space of polynomials of degree n having complex coefficients.
(iii) Let V = L (P2 (C), M 2,2 (C)) and W = L(M 2,3 (C), C2 ).

[2 3 = 6]