Indian Institute of Technology Kharagpur
Department of Mathematics
MA11004 - Linear Algebra, Numerical and Complex Analysis
Problem Sheet - 2
Spring 2022
1. Determine which of the following form a basis of the respective vector spaces:
(a) {4t2 − 2t + 3, 6t2 − t + 4, 8t2 − 8t + 7} of P2 (R).
(b) Let V be a real vector space with {α, β, γ} as a basis. Check whether {α + β + γ, β +
γ, γ} is also a basis of V .
(c)
1 0
0 0
1 1
1 1
1 1
,
,
,
0 0
1 0
1 1
for V , where V is the vector space of all 2 × 2 real matrices.
2. Determine a basis and the dimension of the following subspaces:
(a) The subspace V of all 2 × 2 real symmetric matrices.
(b) U = {(x, y, z, w) ∈ R4 : x + 2y − z = 0, 2x + y + w = 0} of R4 .
Z 1
(c) Let U = {p ∈ P4 (R) :
p(t)dt = 0}.
−1
3. Let S = {(x, y, z, w) ∈ R4 : x+y+z+w = 0} and T = {(x, y, z, w) ∈ R4 : 2x+y−z+w = 0}
be two subspaces of R4 . Find the dimension of S ∩ T .
4. Examine whether T is a linear transformation or not
(i) T : R2 → R2 defined by T (x, y) = (x + y, x − y), (x, y) ∈ R2 .
(ii) T : R2 → R3 defined by T (x, y) = (x + 2y, 2x + y, x + y), (x, y) ∈ R2 .
(iii) T : R3 → R3 defined by T (x, y, z) = (yz, zx, xy), (x, y, z) ∈ R3 .
5. Consider the vector space C over the field C. Give an example of a function φ : C → C
such that φ(w + z) = φ(w) + φ(z) ∀ w, z ∈ C, but φ is not a linear transformation .
6. Find the null space and range space of the following linear transformations. Also find
their respective dimensions and verify the rank-nullity theorem.
Z x
(a) T : P2 (R) → P3 (R) defined by T (f (x)) = 2f 0 (x) +
3f (t)dt.
0
3
2
(b) T : R → R , defined by T (x, y, z) =
( x−y−z
, z2 ).
2
(c) T : M2×2 (R) → M2×2 (R) defined by T (A) =
A − AT
, ∀A ∈ M2×2 (R).
2
7. (a) Determine the linear transformation T : R2 → R3 such that T (1, 1) = (1, 0, 2), T (2, 3) =
(1, −1, 4).
(b) Determine the linear transformation T : R3 → R2 which maps the basis vectors
{(1, 0, 0), (0, 1, 0), (0, 0, 1)} of R3 to the vectors {(1, 1), (2, 3), (3, 2)} respectively.
1
(i) Find T (1, 1, 0), T (6, 0, −1),
(ii) Find N (T ) and R(T ),
(iii) Prove that T is not one-to-one but onto.
8. Find the matrix of the linear transformations with respect to the given ordered bases:
(a) D : P4 (R) → P4 (R) defined by D(p(x)) = 3
basis {1, x, x2 , x3 , x4 } for both P4 (R).
(b) T : P3 (R) → M2×2 (R) by
d3
(p(x)), with respect to the ordered
dx3
00
2f (0) f (3)
T (f (x)) =
0
f 0 (2)
1 0
0 1
0 0
0 0
2 3
with respect to the ordered basis {1, x, x , x } and
,
,
,
.
0 0
0 0
1 0
0 1
9. Determine the linear mapping T : R3 → R4 that maps the basis vector (0,1,1), (1,0,1),
(1,1,0) of R3 to the vectors (0,1,1,1), (1,0,1,1), (1,1,0,1), respectively. Find null space
N (T ) and range space R(T ) and also verify that dim N (T ) + dim R(T ) = 3.
10. Find the rank of the following matrices



1
3 1 −5 1



−2
1 −2 1 −5
(ii)
(i)
−1
1 5 −7 2



1
−1 2 −2
(iv)  2
(iii)  4 −3 4 
3
−2 4 −4
[Answers. (i) 3
(ii) 2
(iii) 2

−2 3
4
4 −1 −3 
2
7
6

2 −1 3
4 1 −2 
6 3 −7
(iv) 2 ]
11. Use Gauss elimination method to conclude about the (in)consistency of the system
x + y − 3z = 4
2x + y − z = 2
3x + 2y − 4z = 7
[Hint. rank A = 2 and rank of the augmented matrix is 3, so inconsistent]
12. Use Gauss elimination method to show that the following system has a unique solution.
Find also the solution.
2x − 2y
= −6
x−y+z = 1
3y − 2z = − 5
13. Show that the system
x + 2y + 4z + w = 4
2x
− z − 3w = 4
x − 2y − z
= 0
3x + y − z − 5w = 5
2
is inconsistent.
[Hint. rank A = 3 and rank of the augmented matrix is 4]
14. Determine the condition for which the system
x+y+z =b
2x + y + 3z = b + 1
5x + 2y + az = b2
admits of (i) only one solution, (ii) no solution, (iii) infinitely many solutions.
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