Exercises for tutorials

EXERCISES IN LINEAR ALGEBRA
1. Matrix operations
(1) Put D = diag(d1 , d2 , . . . , dn ). Let A = (aij ) be an n × n matrix. Find DA and AD.
When is D invertible ?
(2) An n × n matrix A = (aij ) is called upper triangular if aij = 0 whenever i > j. Prove
that product of upper triangular matrices is upper triangular.
(3) Let A be an m × n matrix and B be an n × p matrix. Let Ai denote the ith row of
A and Aj denote the j th column of A. Show that AB = (AB 1 , AB 2 , . . . , AB p ) and
AB = (A1 B, A2 B, . . . , Am B)t .
(4) Prove that a matrix that has a zero row or a zero column is not invertible.
(5) A square matrix A is called nilpotent if Ak = 0 for some positive integer k. Show
that if A is nilpotent then I + A is invertible.
(6) Find infinitely many matrices B such that BA = I2 where


2 3


A =  1 2 .
2 5
(7)
(8)
(9)
(10)
(11)
(12)
(13)
Show that there is no matrix C such that AC = I3 .
Let A and B be square matrices. Let tr(A) denote the trace of A which is the sum of
its diagonal entries. Show that for two n × n matrices A and B, tr(A + B) = tr(A) +
tr(B) and tr(AB) = tr(BA). Show that if B is invertible then tr(A) = tr(BAB −1 ).
Show that the equation AB − BA = I has no solutions in Rn×n .
Show that for any matrix A, AAt is symmetric. Show that every square matrix is
uniquely a sum of a symmetric and skew-symmetric matrix.
Show that every matrix in Cn×n is uniquely a sum of a Hermitian and skew-Hermitian
matrix.
Show that inverse of an invertible symmetric matrix is also symmetric.
Consider a system of linear equations Ax = b where A ∈ Rm×n , x = (x1 , x2 , . . . , xn )t
and b ∈ Rm .
(a) Show that if Ax = b has more than one solution then it has infinitely many.
(b) Prove that if there is a complex solution then there is a real solution.
Find all 2 × 2 matrices A such that A2 = −I.
1
2
(14) Find all 2 × 2 matrices A such that A2 = 0.
(15) Show that if A3 − A + I = 0 then A is invertible.
2. Vector spaces, subspaces, basis and dimension
(16) Determine which of the following subsets of Rn are subspaces ?
(a) V1 = {(x1 , x2 , . . . , xn ) | x1 = 1}.
(b) V2 = {(x1 , x2 , . . . , xn ) | x1 = 0}.
P
(c) V3 = {(x1 , x2 , . . . , xn ) | ni=1 xi yi = 0}. Here (y1 , y2 , . . . , yn ) ∈ Rn } is a fixed
vector.
(17) Let V = C(R) = {f : R → R | f is continuous}. Determine which of the following
are subspaces of C(R).
(a) V4 = {f ∈ V | f (1/2) is a rational number}.
R1
(b) V5 = {f ∈ V | 0 f (t)dt = 0}.
2
(c) V5 = {f ∈ V | a ddt2f + b df
+ cf = 0}. Here a, b, c ∈ R are fixed.
dt
(18) Find a basis of the subspace of Rn of the solutions of the equation x1 +x2 +· · ·+xn = 0.
(19) Show that the vector space F [x] of all polynomials over a field F is not finitely
generated.
(20) Is the vector space C(R) finite dimensional ?
(21) Show that the set {1, (x − a), (x − a)2 , (x − a)3 , . . . , (x − a)n } for a fixed a ∈ F is
a basis of the vector space Pn (F ) of all polynomials with coefficients in F of degree
atmost n.
(22) Let u1 , u2 , . . . un be linearly independent vectors in a vector space V. Show that any
vector in L(u1 , u2 , . . . , un ) is unique linear combination of u1 , u2 , . . . , un .
(23) Describe all the subspaces of R3 .
(24) Let U and V be subspaces of a vector space W. Suppose that U ∩ V = (0) and
dim W = dim U + dim V. Show that any w ∈ W there exist unique vectors u ∈ U
and v ∈ V such that w = u + v.
(25) What is the dimension of the Q-vector space R ?
(26) Determine whether (1, 1, 1) ∈ L{(1, 3, 4), (4, 0, 1), (3, 1, 2)}.
(27) Prove that every subspace W of a finitely generated vector space V is finitely generated. Prove that dim W ≤ dim V with equality if and only if V = W.
(28) Let F be a field with two elements. Let V be a two dimensional vector space over F.
Count the number of elements of V, the number of subspaces of V and the number
of different bases.
(29) Let S and T be two dimensional subspaces of R3 . Show that dim(S ∩ T ) ≥ 1.
(30) Find bases of the following vector spaces: (a) the vector space of all n × n real upper
triangular matrices, (b) the vector space of all real n × n symmetric matrices, (c)
3
the vector space of all real n × n skew-symmetric matrices and (d) the vector space
of all homogeneous polynomials of degree d in n variables together with the zero
polynomial.
3. Systems of linear equations, rank of a matrix
(31) Test for solvability of the following systems of equations, and if solvable, find all the
solutions.
(a)
x1 + x2 + x3 = 8
x1 + x2 + x4 = 1
x1 + x3 + x4 = 14
x2 + x3 + x4 = 14
(b)
x1 + 2x2 + 4x3 = 1
2x1 + x2 + 5x3 = 0
3x1 − x2 + 5x3 = 0
(32) For what vaules of a does the following system of equations have a solution ?
3x1 − x2 + ax3 = 1
3x1 − x2 + x3 = 5
(33) Prove that a system of m homogeneous linear equations in n > m unknowns always
has a nontrivial solution.
(34) Show that a system of homogeneous linear equations in n unknowns has a nontrivial
solution if and only if the coefficient matrix has rank less than n.
(35) Find a basis of the solution space of the system
3x1 − x2 + x4 = 0
x1 + x2 + x3 + x4 = 0
(36) Find a point in R3 where the line joining the points (1, −1, 0) and (−2, 1, 1) pierces
the plane 3x1 − x2 + x3 − 1 = 0.
(37) Using row and column operations find the rank of the matrix


1
2 −3
 −1 −2
3 




 4
8 −12 
1 −1
5
4
(38) Find the rank of an upper triangular matrix in terms of the diagonal entries.
(39) Let A be an m × n matrix and B be an n × r matrix.
(a) Show that the columns of AB are linear combinations of the columns A. Hence
prove that rank(AB) ≤ rank(A).
(b) Using (a) and the fact that rank of a matrix and its transpose are equal, prove
that rank(AB) ≤ rank(B).
(40) Let A be an n × n matrix such that rank(A) = rank(A2 ). Find all the vectors in the
column space of A which are solutions to Ax = 0.
4. Linear Transformations
(41) Let F be a field and F n denote the vector space F n×1 . Let T : F 2 → F 2 be the
linear transformation T ((x, y)t ) = (−3x + y, x − y)t . Let U : F 2 → F 2 be the linear
transformation U ((x, y)t ) = (x + y, x)t . Describe the linear transformations U T, T U
and T 2 + U. Is T U = U T ?
(42) Test whether the linear transformations T : R2 → R2 , T ((x1 , x2 )t ) = (y1 , y2 )t defined
below are one-to-one.
(a) y1 = 3x1 − x2 , y2 = x1 + x2 .
(b) y1 = x1 + 2x2 + x3 , y2 = x1 + x2 , y3 = x2 + x3 .
Rx
(43) Let I : R[x] → R[x] be the linear map I(f (x)) = 0 f (x)dx. Let D : R[x] → R[x]
be the linear map D(f (x)) = f 0 (x). Show that DI = I but neither D nor I are
isomorphisms. Is D (resp. I) one-to-one or onto ? Find the ranks and nullities of D
and I.
(44) Let S, T : R2 → R2 be the linear maps defined by the equations
S(u1 ) = u1 − u2 , S(u2 ) = u1 and T (u1 ) = u2 , T (u2 ) = u1 ,
where B = {u1 , u2 } is a basis of R2 . Let C = {w1 = 3u1 − u2 , w2 = u1 + u2 }.
Show that C is a basis of R2 . Find the matrices MBB (S), MBB (T ), MCC (S), MCC (S).
Find invertible matrices X in each case such that X −1 AX = A0 where A is the
matrix of the transformation with respect to the old basis and A0 is the matrix of the
transformation with respect to the new basis.
(45) Let B = {u1 , u2 } be a basis of R2 . Let S and T be the linear maps defined by the
equations
S(u1 ) = u1 + u2 , S(u2 ) = −u1 − u2 and T (u1 ) = u1 − u2 , T (u2 ) = 2u2 .
Find the rank and nullity of S and T. Which of these linear maps are invertible ?
(46) Let V be a vector space of dimension n and T : V → V be a linear map. Let A be
the matrix of T with respect to any basis of V. Show that rank(T ) = rank(A).
5
(47) Let V be an n-dimensional vector space. Let T : V → V be a linear transformation
such that the nullspace and the range of T are same. Show that n is even. Give an
example of such a map for n = 2.
(48) Let T be the linear operator on R3 defined by the equations:
T ((x1 , x2 , x3 )t ) = (3x1 , x1 − x2 , 2x1 + x2 + x3 )t .
Is T invertible ? If so, find a formula for T −1 .
(49) Let V = C2×2 be the vector space of 2 × 2 complex matrices. Let
"
#
1 −1
B=
.
−4
4
Define T : V → V by T (A) = BA. Find rank of T . Describe T 2 .
(50) Let V be vector space with dim V = n and T : V → V be a linear map such that
rank T 2 = rank T. Show that N (T ) ∩ T (V ) = (0). Give an example of such a map.
(51) Let T be a linear operator on a finite-dimensional vector space V. Suppose that U is
a linear operator on V such that T U = I. Prove that T is invertible and U = T −1 .
(52) Let W be the real vector space all 2 × 2 complex Hermitian matrices. Show that the
map
"
#
t
+
x
y
+
iz
(x, y, z, t)t 7→
y − iz t − x
(53)
(54)
(55)
(56)
is an isomorphism of R4 onto W.
Let V and W be finite dimensional vector spaces over a field F. Show that V is
isomorphic to W if and only if dim V = dim W.
Show that every matrix A ∈ F m×n of rank one has the form A = uv t where u ∈ F m×1
and v ∈ F n×1 .
Let V be the infinite-dimensional real vector space of all real sequences. Let R :
V → V be the right shift operator R((a1 , a2 , . . .)) = (0, a1 , a2 , . . .) and L : V → V
denote the left shift operator defined by L((a1 , a2 , . . .)) = (a2 , a3 , . . .). Show that R is
one-to-one but not onto and L is onto but not one-to-one.
Let V be an n-dimensional vector space over a field F. Let B = {u1 , u2 , . . . , un }
be a basis of V. Define the linear transformation T : V → V by T (ui ) = ui+1 for
i = 1, 2, . . . , n − 1 and T (un ) = 0.
(a) Find the matrix A = MBB (T ).
(b) Show that T n−1 6= 0 but T n = 0.
(c) Let S be any linear operator on V such that S n = 0 but S n−1 6= 0. Prove that
there is a basis C of V such that MCC (S) = A.
(d) Let M, N ∈ F n×n and M n = N n = 0 but M n−1 6= 0 and N n−1 6= 0. Show that
M and N are similar matrices.
6
(57) Let V be a two-dimensional vector space over a field F. Let T be a linear operator
on V so that the matrix of T with respect to a basis B of V is
"
#
a b
.
c d
Show that T 2 − (a + d)T + (ad − bc)I = 0.
(58) Let T : V → W be a linear map of vector spaces. Suppose V is infinite-dimensional.
Prove that at least one of N (T ) or T (V ) is infinite-dimensional.
(59) Let V be the vector space of all real functions continuous on [a, b]. Define T : V → V
by the equation
Z b
T (f (x)) =
f (t) sin(x − t)dt for a ≤ x ≤ b.
a
Find the rank and nullity of T.
(60) Let V denote the vector space of all real functions continuous on the interval [−π, π].
Let S denote the subspace of V consisting of all f satisfying
Z π
Z π
Z π
f (t) sin tdt = 0 .
f (t) cos tdt = 0,
f (t)dt = 0,
−π
−π
−π
Prove that S contains the functions f (x) = sin nx and f (x) = cos nx for all n =
2, 3, . . . . Show that S is infinite-dimensional.
5. Inner product spaces
(61) Let V be the subspace spanned by the vectors u1 = (−1, 1, 1, 1), u2 = (1, −1, 1, 1), u3 =
(1, 1, −1, 1) in R4 . Find an orthonormal basis of V by Gram-Schmidt process.
(62) Find an orthonormal basis of the vector space V = P3 (R) of all real polynomials of
R1
degree atmost 3 with the inner product < f, g >= 0 f (t)g(t)dt. Take {1, x, x2 , x3 }
as a basis of V.
(63) Let V = C[−π, π] be the vector space of all continuous real valued functions defined
on the interval [−π, π]. Then V is an inner product space with the inner product
Rπ
< f, g >= −π f (t)g(t)dt. Show that the functions 1, sin nx, cos nx, n = 1, 2, . . . form
an orthogonal set.
(64) Two vector spaces V and W with inner products < v1 , v2 > and [w1 , w2 ] respectively, are said to be isometric if there is an isomorphism T : V → W such that
[T (v1 ), T (v2 )] =< v1 , v2 > for all v1 , v2 ∈ V. Such a T is called an isometry. Let V be
a finite-dimensional inner product space over a field F with inner product < u, v > .
Let B = {v1 , v2 , . . . , vn } be an orthonormal basis of V. Let T : V → F n be the linear
map T (v) = MB (v). Consider F n with standard inner product. Show that T is an
isometry.
7
(65) Let V be a finite-dimensional inner product space. Let W be a subspace. Show
that the set W ⊥ = {v ∈ V | < v, w >= 0 for all w ∈ W } is a subspace of V and
dim W + dim W ⊥ = dim V.
(66) Show that every subspace of Cn with standard inner product is the subspace of all
solutions to a system of homogeneous linear equations.
(67) Let A = (aij ) ∈ R2×2 . For u, v ∈ R2 define fA (u, v) = v t Au. Show that fA is an inner
product on R2 if and only if A = At , a11 > 0, a22 > 0 and det A > 0.
(68) Let V = Cn×n with the inner product < A, B >= tr(AB ∗ ). Let D be the subspace
of diagonal matrices. Find D⊥ .
(69) Let W be a finite-dimensional subspace of an inner product space V. Let E be the
orthogonal projection of V onto W. Prove that < Eu, v >=< u, Eu > for all u, v ∈ V.
(70) Let V be a finite-dimensional inner product space and let B = {u1 , u2 , . . . , un } be
a basis of V. Let T : V → V be a linear map. Put MBB (T ) = (aij ). Show that
aij =< T uj , ui > .
(71) Let A be a symmetric n × n real matrix. Let u, v ∈ V = Rn×1 \ {0} and λ, µ ∈ R
such that Au = λu and Av = µv. Show that u ⊥ v. In other words, eigenvectors for
distinct eigenvalues of symmetric real matrices are orthogonal.
(72) Let B = {u1 , u2 , . . . , un } be an orthonormal set of vectors in an inner product space
P
V. Show that for any v ∈ V, ni=1 |hv, ui i|2 ≤ ||v||2 and equality holds if and only if
v ∈ L(B).
(73) Let p, q, r ∈ Z. Show that the vectors (p, q, r)t , (q, r, p)t , (r, p, q)t ∈ R3 are mutually
orthogonal if and only if pq + qr + rp = 0. Show that in this case, the length of each
of these vectors in |p + q + r|.
(74) Let U, V be subspaces of an inner product space W. Let dim U < dim V. Show that
there is a nonzero vector in V lying in U ⊥ .
(75) Without using Gram-Schmidt orthogonalization process, find the orthogonal projection of (1, 2, 2, 9)t ∈ R4 in the column space of the matrix



A=

2
1
−1
2
1
2
0
1



.

8
6. Determinants
(76) Find the inverses of the following matrices by Gauss-Jordan method and the adjoint
formula:


"
#
3
1 0
a b


, ad − bc 6= 0  1
2 1 .
c d
0 −1 2
(77) Find the ranks of the following matrices by using determinants:


"
#
−1 0 1 2
1 2 3 4


,  1 1 3 0 .
−1 2 1 0
−1 2 4 1
(78) Show that the equation of line through
given by
x y 1
a b 1
c d 1
the distinct points (a, b) and (c, d) in R2 is
= 0.
(79) Show that the equation of the plane in R3
(a, b, c); (d, e, f ); (g, h, k) is given by
x y z 1
a b c 1
d e f 1
g h k 1
passing through three non-collinear points
= 0.
(80) Show that the area of the triangle with vertices (a, b); (c, d); (e, f ) in the plane is given
by the absolute value of
a b 1 1
c d 1 .
6
e f 1 (81) Show that the volume of the tetrahedron with vertices (a1 , a2 , a3 ), (b1 , b2 , b3 ), (c1 , c2 , c3 ),
(d1 , d2 , d3 ) is given by the absolute value of

a1 a2 a3 1
1 b1 b2 b3 1 

.
6 c1 c2 c3 1 
d1 d2 d3 1
9
(82) Prove the following formula for the van der Monde determinant:
1 a a2 . . . an−1 1
1
1
n−1 2
1 a2 a2 . . . a2 Y
=
(aj − ai ).
Vn = . .
..
..
..
.
.
.
.. ..
i<j
1 an a2n . . . an−1
n
(83)
(84)
(85)
(86)
(87)
(88)
(89)
(90)
[Hint: Use induction on n. Multiply each column by a1 and subtract it from the
next column on the right, starting from the right hand side. Prove that Vn = (an −
a1 )(an−1 − a1 ) . . . (a2 − a1 )Vn−1 . ]
If A ∈ Cn×n is a skew-symmetric matrix where n is odd, then show that det A = 0.
Let A be an orthogonal matrix, that is, AAt = I. Show that for such a matrix,
det A = ±1. Give an example of an orthogonal matrix with determinant −1.
A complex n × n matrix is called unitary if AA∗ = I. Here A∗ denotes the conjugate
transpose of A. If A is unitary, show that | det A| = 1.
Let V = F n×n . Let B ∈ V. Define TB : V → V by TB (A) = AB − BA. Show that
det TB = 0.
Let A, B ∈ F n×n where A is invertible. Show that there are atmost n scalars for
which cA + B is not invertible.
Let V = F n×n and B ∈ V. Define LB , RB : V → V by LB (A) = BA and RB (A) = AB
for all A ∈ V. Show that det RB = det LB = (det B)n .
Let V = F 1×n and let T : V → V be a linear operator. Define f (u1 , u2 , . . . , un ) =
det(T u1 , T u2 , . . . , T un ). (1) Show that f is multilinear and alternating. (2) Let B be
any ordered basis of V and A = [T ]B . Show that det A = det T = f (e1 , e2 , . . . , en ).
Let B ∈ V = Cn×n . Define the linear operator MB : V → V by MB (A) = BAB ∗ .
Show that det MB = | det B|2n .
7. Diagonalization of matrices and operators
Let V be an n-dimensional vector space over a field F and T be a linear operator on
V in the following problems unless stated otherwise.
Eigenvalues and eigenvectors
(91) Show that similar matrices have same characteristic polynomials and hence have same
eigenvalues and traces.
(92) Find the eigenvalues and eigenspaces of the following matrices and determine if they
are diagonalizable:
#
# "
"
1
0
4
5
.
,
1 −2
−1 −2
10
(93) Let f (x) ∈ F [x]. Show that α is an eigenvalue of T if and only if f (α) is an eigenvalue
of f (T ).
(94) Let A, B ∈ F n×n . Prove that if I − AB is invertible then I − BA is invertible and
(I − BA)−1 = I + B(I − AB)−1 A.
(95) Use the result of the above exercise to show that AB and BA have the same characteristic polynomials.
(96) Let T be an invertible linear operator on a vector space V. Show that λ is an eigenvalue
of T if and only if λ 6= 0 and λ−1 is an eigenvalue of T −1 .
(97) Let N ∈ C2×2 and N 2 = 0. Prove that either N = 0 or N is similar over C to
"
#
0 0
.
1 0
(98) Let A ∈ C2×2 . Show that A is similar over C to a matrix of one of the two types:
"
# "
#
a 0
a 0
,
.
0 b
1 a
(99) Let V denote the vector space of all continuous real valued functions defined on R.
Rx
Let T be the linear operator T (f (x)) = 0 f (t)dt. Show that T has no eigenvalues.
(100) Let A be an n × n diagonal matrix with characteristic polynomial
CA (x) = (x − c1 )d1 (x − c2 )d2 . . . (x − ck )dk .
where c1 , c2 , . . . , ck are distinct. Let V be the vector space of all n × n matrices B
such that AB = BA. Prove that dim V = d21 + d22 + · · · + d2k .
Minimal polynomials
(101) Find the minimal polynomials of

 

"
#
0 1 0
0 1 3
2
0

 

,  0 0 1 ,  0 0 2 .
3 −1
1 0 0
0 0 0
(102) Find an n × n nilpotent matrix with minimal polynomial x2 .
(103) Show that the following matrices have the same minimal polynomials:

 

2 0 0
0
−1
0 0
0

 0 −1 0
0 
0 

  0 2 0

.
, 

 0
0 
0 2
0   0 0 2
0 0 0 −1
0
0 0 −1
(104) Prove that a linear operator T defined on a finite dimensional vector space V is
invertible if and only if its minimal polynomial has a nonzero constant term. Describe
how to find T −1 from its minimal polynomial.
(105) Let T be a nilpotent operator on V. Show that T n = 0.
11
(106) Let V = F n×n . Let A ∈ V be a fixed matrix. Let T be the linear operator on V
defined by T (B) = AB. Show that T and A have same minimal polynomial.
(107) Let m(x) be the minimal polynomial of T and f (x) ∈ F [x]. Let d(x) be the greatest
common divisor of m(x) and f (x). Show that the nullspaces of f (T ) and d(T ) are
equal.
Diagonalization
(108) Show that the matrix


0 0 1


 1 0 0 
0 1 0
(109)
(110)
(111)
(112)
(113)
(114)
is similar to a diagonal matrix in C3×3 but not in R3×3 .
Let T have n distinct eigenvalues. Show that T is diagonalizable.
Show that every matrix A such that A2 = A is diagonalizable.
Show that the orthogonal projection operators are diagonalizable.
When is a nilpotent operator diagonalizable ?
Show that the differentiation operator defined on the space of real polynomials of
degree atmost n ≥ 1 is not diagonalizable.
Let T : R4 → R4 be the linear operator induced by the matrix


0 0 0 0
 a 0 0 0 



.
 0 b 0 0 
0 0 c 0
Find necessary and sufficient conditions on a, b, c so that T is diagonalizable.
(115) Show that a 2 × 2 real symmetric matrix is diagonalizable.
8. Projections and invariant direct sums
Projections
(116) Find a projection E : R2 → R2 so that E(R2 ) is the subspace spanned by (1, −1)t
and N (E) is spanned by (1, 2)t .
(117) Let E1 and E2 be projections onto independent subspaces of a vector space V. Is
E1 + E2 a projection ?
(118) Is it true that a diagonalizable operator with only eigenvalues 0 and 1 is a projection
?
(119) Let E : V → V be a projection. Show that I − E is a projection along E(V ) onto
N (E).
12
(120) Let V be a real vector space and let E : V → V be a projection. Prove that I + E
is invertible and find its inverse.
(121) Let F be a subfield of C. Let E1 , E2 , . . . , Ek be projections of an n-dimensional F vector space V such that E1 + E2 + · · · + Ek = I. Prove that Ei Ej = 0 for i 6= j.
[Hint: Use the fact that the trace of projection is its rank.]
Invariant subspaces and direct sums
(122) Let E be a projection of V and T ∈ L(V, V ). Prove that E(V ) is invariant under T
if and only if ET E = T E. Prove that E(V ) and N (E) are invariant under T if and
only if T E = ET.
(123) Let T : R2 → R2 be the linear operator induced by the matrix
#
"
2 1
.
0 2
Let W1 = L(e1 ). Prove that W1 is invariant under T. Show that there is no subspace
W2 of R2 that is invariant under T and R2 = W1 ⊕ W2 .
(124) Let T be a linear operator on V. Suppose V = W1 ⊕ W2 ⊕ · · · ⊕ Wk where each Wi
is invariant under T. Let Ti be the restriction of T on Wi .
(a) Prove that det T = det T1 det T2 . . . det Tk .
(b) Show that CT (x) = CT1 (x)CT2 (x) . . . CTk (x).
(c) Show that the minimal polynomial of T is the least common multiple of the
minimal polynomials of T1 , T2 , . . . , Tk .
(125) Let T be the linear operator on V = R3 induced by the matrix


5 −6 −6


A =  −1
4
2 .
3 −6 −4
Use Lagrange polynomials to find matrices E1 , E2 ∈ R3×3 so that A = E1 +2E2 , E1 +
E2 = I and E1 E2 = 0.
(126) Let T be a linear operator on V which commutes with every projection operator on
V. What can you say about T ?
9. Primary decomposition, cyclic subspaces and Jordan form
(127) Let T be a linear operator on the finite-dimensional vector space V with charQk
di
acteristic polynomial cT (x) =
and minimal polynomial mT (x) =
i=1 (x − ci )
Qk
ri
ri
m
i=1 (x − ci ) . Let Wi = Null(T − ci I) . Show Wi = {u ∈ V | (T − ci I) u =
0 for some m depending on u}. and dim Wi = di .
13
(128) Let T be a rank one linear operator on a finite dimensional vector space V. Show that
either T is diagonalizable or T is nilpotent but not both.
(129) Show that two 3 × 3 nilpotent matrices over a field F are similar if and only if they
have same minimal poynomials.
(130) Give an example of two 4×4 nilpotent matrices which have same minimal polynomials
but which are not similar.
(131) Let T be the linear map induced on R3 by the matrix diag(2, 2, −1). Show that T
has no cyclic vector.
(132) Let T be a diagonalizable linear operator on an n-dimensional vector space V.
(a) If T has a cyclic vector, show that the characteristic polynomial has n distinct
roots.
(b) If T has n distinct eigenvalues and {u1 , u2 , . . . , un } is a basis of eigenvectors then
u = u1 + u2 + · · · + un is a cyclic vector for T.
(133) Let A be a complex 5×5 matrix with characteristic polynomial f (x) = (x−2)3 (x+7)2
and minimal polynomial p(x) = (x − 2)2 (x + 7). What is the Jordan form of A ?
(134) Let V be the complex vector space of polynomials of degree atmost 3. Let D : V → V
be the differentiation operator. Find the Jordan form of the matrix of D in the
standard basis B = {1, x, x2 , x3 } of V.
(135) Let N be a k × k nilpotent matrix whose degree of nilpotency is k. Show that N t is
similar to N. Show that every complex n × n matrix is similar to its transpose.
(136) Let N ∈ V = F n×n be a nonzero nilpotent matrix with N n = 0 but N n−1 6= 0. Show
that there is no matrix A ∈ V such that A2 = N.
(137) Let N be a 3 × 3 complex nilpotent matrix. Prove that A = I + 21 N − 18 N 2 satisfies
A2 = I + N. We say that A is a square root of I + N. Let N be any n × n complex
nilpotent matrix. Find a formula for square root of I + N.
(138) Use Jordan form to prove that every invertible n × n complex matrix has a square
root.
(139) Find the Jordan canonical form over C of the matrices:


#
"
0 0 1
2 −1


 1 0 0 .
1 −1
0 1 0
(140) Let A ∈ Rn×n such that A2 + I = 0. Prove that n = 2k for some k ∈ N and A is
similar over R to the matrix in block form:
"
#
0 −I
I
0
where I is the k × k identity matrix.
14
10. Spectral Theory and its applications
Linear functionals and adjoints
(141) Let V be a finite dimensional inner product space and T a linear operator on V. Show
that T ∗ (V ) = Null(T )⊥ .
(142) Let V be an inner product space and v, w ∈ V be fixed vectors. Define T (u) =
hu, viw. Show that T has an adjoint and find T ∗ . Now let V = Cn . Find the rank of
T and the matrix of T in standard basis of Cn .
(143) Let V be the vector space of real polynomials of degree atmost 3, with the inner
R1
product hf (t), g(t)i = 0 f (t)g(t)dt. Let r ∈ R and let T (f (t)) = f (r). Show that
T is a linear functional and find g(t) ∈ V such that T (f (t)) = hf (t), g(t)i for all
f (t) ∈ V.
(144) Let D be the differentiation operator on V as defined in the problem 143. Find D∗ .
(145) Let V = Cn×n with inner product hA, Bi = tr(AB ∗ ). Let P ∈ V be a fixed invertible
matrix and define T (A) = P −1 AP for all A ∈ V. Find the adjoint of T.
(146) Let T be a linear operator on a finite dimensional inner product space V. Show that
T is self-adjoint if and only if hT u, ui ∈ R for all u ∈ V.
Unitary operators
(147) Let V be as in problem 145. For a fixed M ∈ V, Define T (A) = M A. Show that T is
unitary if and only if M is a unitary matrix.
(148) Let V = R2 with standard inner product. Let U be a unitary operator on V. Let E
be the standard basis of V. Show that
#
"
#
"
cos θ − sin θ
cos θ
sin θ
[U ]E = Uθ =
or
.
sin θ
cos θ
sin θ − cos θ
Find the adjoint of Uθ .
(149) Let V = R2 with standard inner product. Let W be the plane spanned by u =
(1, 1, 1)t and v = (1, 1, −2)t . Let U be the anticlockwise rotation through an angle
θ about the line perpendicular to W when seen from a high point above the plane.
Find the matrix of U in the standard basis of R2 .
(150) Let V be a finite dimensional inner product space and W a subspace of V. Then
V = W ⊕ W ⊥ . Let u = v + w where u ∈ V, v ∈ W, and w ∈ W ⊥ . Define U (u) =
v − w. Prove that U is self-adjoint and unitary. Prove that every self-adjoint unitary
operator on V arises this way from a subspace W of V. Let W be the linear span of
(1, 0, 1)t . Find the matrix of U in the standard basis.
15
(151) Let V be an inner product space. A function T : V → V is called a rigid motion if
||T (u) − T (v)|| = ||u − v|| for all u, v ∈ V.
(a) Show that unitary operators are rigid motions.
(b) Let w ∈ V be fixed. Define Tw , translation by w, by Tw (u) = u + w for all u ∈ V.
Show that translations are rigid motions.
(c) Let V = R2 . Let T be a rigid motion of V with T (0) = 0. Show that T is linear
and unitary.
(d) Show that every rigid motion of R2 is translation followed by a rotation or a
reflection and a rotation.
Normal operators
(152) For


1 2 3


A= 2 3 4 
3 4 5
find an orthogonal matrix P such that P t AP is a diagonal matrix.
(153) Find an orthonormal basis of C2 with standard inner product consisting of eigenvectors of the normal matrix
"
#
1 i
A=
.
i 1
(154) A linear operator T on an inner product space V is called positive if T is self-adjoint
and hT u, ui is a positive real number. Show that a normal linear operator T on a
finite-dimensional inner product space V is self-adjoint, positive or unitary according
as every eigenvalue of T is real, or positive or has absolute value 1. Find all positive
and unitary operators on V.
(155) Show that a linear operator T defined on an inner product space V is normal if
and only if T = T1 + iT2 where T1 and T2 are self-adjoint operators on V such that
T1 T2 = T2 T1 .
(156) Show that a real symmetric matrix has a real symmetric cube root.
(157) Let T be a normal operator on a finite dimensional inner product space. Show that
there is a complex polynomial f (x) such that T ∗ = f (T ).
(158) Let A ∈ Rn×n be a symmetric matrix with Ak = I for some k. Show that A2 = I.
P
(159) Show that if a normal linear operator T has spectral decomposition T = ki=1 ai Ei
then for any polynomial f (x) ∈ C[x], the spectral decomposition of f (T ) is given by
P
f (T ) = ki=1 f (ai )Ei .
(160) Using spectral theorem for symmetric matrices draw the conic section 9x2 + 24xy +
16y 2 −20x+15y = 0 and the quadric surface 7x2 +7y 2 −2z 2 +20yz −20zx−2xy = 36.