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IITM BS Week 6 Mathematics 2 Questions | Prasnya
Mathematics 2 > Week 6
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38Q
01
Let
T
:
R
3
→
R
2
T: \mathbb{R}^3 \to \mathbb{R}^2
T
:
R
3
→
R
2
be a linear transformation. Let
B
1
=
{
(
1
,
0
,
1
)
,
(
1
,
1
,
0
)
,
(
0
,
1
,
1
)
}
B_1 = \{(1, 0, 1), (1, 1, 0), (0, 1, 1)\}
B
1
=
{(
1
,
0
,
1
)
,
(
1
,
1
,
0
)
,
(
0
,
1
,
1
)}
and
B
2
=
{
(
1
,
1
)
,
(
1
,
−
1
)
}
B_2 = \{(1, 1), (1, -1)\}
B
2
=
{(
1
,
1
)
,
(
1
,
−
1
)}
be ordered bases for the domain and co-domain respectively. Suppose that the matrix representation of
T
T
T
with respect to
B
1
B_1
B
1
and
B
2
B_2
B
2
is
[
T
]
B
1
B
2
=
[
0
−
1
2
−
1
2
0
]
[T]_{B_1}^{B_2} = \begin{bmatrix} 0 & -1 & 2 \\ -1 & 2 & 0 \end{bmatrix}
[
T
]
B
1
B
2
=
[
0
−
1
−
1
2
2
0
]
. Which of the following defines the transformation
T
T
T
?
Single correct
HARD
6 marks
10 May 2026
02
Let
T
:
R
3
→
R
2
T: \mathbb{R}^3 \to \mathbb{R}^2
T
:
R
3
→
R
2
be a linear transformation defined by
T
(
x
,
y
,
z
)
=
(
x
−
2
y
,
y
−
3
z
)
T(x, y, z) = (x - 2y, y - 3z)
T
(
x
,
y
,
z
)
=
(
x
−
2
y
,
y
−
3
z
)
. Let
B
1
=
{
(
1
,
0
,
1
)
,
(
1
,
1
,
0
)
,
(
0
,
1
,
1
)
}
B_1 = \{(1, 0, 1), (1, 1, 0), (0, 1, 1)\}
B
1
=
{(
1
,
0
,
1
)
,
(
1
,
1
,
0
)
,
(
0
,
1
,
1
)}
and
B
2
=
{
(
1
,
1
)
,
(
1
,
−
1
)
}
B_2 = \{(1, 1), (1, -1)\}
B
2
=
{(
1
,
1
)
,
(
1
,
−
1
)}
be ordered bases. Choose the correct option for the matrix representation of
T
T
T
with respect to ordered bases
B
1
B_1
B
1
for the domain and
B
2
B_2
B
2
for the co-domain.
Single correct
MEDIUM
5 marks
10 May 2026
03
Choose all the correct answers.
Comprehension
MEDIUM
7 marks
21 December 2025
04
Suppose
n
n
n
denotes the nullity of
T
T
T
and
m
m
m
denotes the dimension of the subspace of
M
2
×
2
M_{2 \times 2}
M
2
×
2
spanned by
{
[
1
0
−
1
0
]
,
[
1
−
1
0
0
]
,
[
0
−
1
1
0
]
}
\left\{ \begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \right\}
{
[
1
−
1
0
0
]
,
[
1
0
−
1
0
]
,
[
0
1
−
1
0
]
}
. Then, what is the value of
n
−
m
n - m
n
−
m
?
Comprehension
MEDIUM
5 marks
21 December 2025
05
Choose all the correct answers.
Comprehension
MEDIUM
7 marks
21 December 2025
06
Define a linear transformation S:
R
2
\mathbb{R}^2
R
2
-> M_{2 x 2} by S(a, b) =
[
a
a
b
b
]
\begin{bmatrix} a & a \\ b & b \end{bmatrix}
[
a
b
a
b
]
. Consider the following statements. 1) S o T is an isomorphism. 2) T o S is an isomorphism. 3) S o T is injective but not surjective. Let n denote the number of correct statements and n = 0, if no statement is correct. What is n?
Comprehension
HARD
5 marks
21 December 2025
07
Let
A
A
A
be a
3
×
3
3 \times 3
3
×
3
matrix such that
v
1
,
v
2
,
v
3
v_1, v_2, v_3
v
1
,
v
2
,
v
3
are the columns of
A
A
A
. If
det
(
A
)
=
0
\det(A) = 0
det
(
A
)
=
0
, choose the correct statement(s).
Multiple correct
HARD
3 marks
13 April 2025
08
Suppose A is the matrix representation of T with respect to the standard basis for both domain and co-domain. Then A is given by
Comprehension
EASY
1 marks
13 April 2025
09
Choose the correct option for T.
Comprehension
MEDIUM
1 marks
13 April 2025
10
Choose the correct definition for T.
Comprehension
EASY
1 marks
22 December 2024
11
If B is the matrix representation of T with respect to the ordered basis {(1, 2, 0),
(
0
,
−
1
,
3
)
(0, -1, 3)
(
0
,
−
1
,
3
)
, (2, 3, 1)} for both the domain and the codomain, find the rank of B.
Comprehension
MEDIUM
1 marks
22 December 2024
12
Let A be the matrix representation of T with respect to some ordered basis for both the domain and the codomain. Which of the following options is correct?
Comprehension
MEDIUM
3 marks
22 December 2024
13
The transformation T is
Comprehension
EASY
1 marks
22 December 2024
14
What is the matrix representation of T?
Comprehension
EASY
1 marks
01 September 2024
15
Select all true statements.
Comprehension
MEDIUM
1 marks
01 September 2024
16
T^-1 is a linear transformation that is the inverse of T. Find the matrix representation of T^-1.
Comprehension
MEDIUM
2 marks
01 September 2024
17
Let
T
:
R
5
→
R
3
T: \mathbb{R}^5 \to \mathbb{R}^3
T
:
R
5
→
R
3
be the linear transformation whose matrix representation is
A
A
A
with respect to the standard basis for both domain and co-domain. Which of the following is true?
Comprehension
MEDIUM
1 marks
28 April 2024
18
Which of the following is the matrix representation of T with respect to the basis {(1,1), (-1,1)} for the domain and the standard ordered basis for the co-domain?
Comprehension
MEDIUM
1 marks
28 April 2024
19
Which of the following are valid matrix representations of T with respect to suitable bases for the domain and co-domain? Note that the bases could be different for the domain and co-domain.
Comprehension
HARD
1 marks
28 April 2024
20
Find
Rank
(
A
)
\operatorname{Rank}(A)
Rank
(
A
)
.
Comprehension
EASY
1 marks
24 December 2023
21
Find the Nullity of T
Comprehension
EASY
1 marks
24 December 2023
22
Find the Rank of T
Comprehension
EASY
1 marks
24 December 2023
23
Choose the correct option.
Comprehension
EASY
1 marks
24 December 2023
24
Choose the appropriate linear transformation T for the matrix A.
Comprehension
HARD
3 marks
03 September 2023
25
Let B be the matrix corresponding to the linear transformation T with respect to the basis {(1, 0, 0),
(
0
,
1
,
0
)
(0, 1, 0)
(
0
,
1
,
0
)
, (0, 0, 1)} for both domain and codomain. Which of the following statement(s) is/are true?
Comprehension
MEDIUM
3 marks
03 September 2023
26
Which of the following statement(s) about T is/are true?
Comprehension
MEDIUM
2 marks
03 September 2023
27
Which of the following matrix is A?
Comprehension
MEDIUM
2 marks
30 April 2023
28
Which of the following is/are true about T?
Comprehension
MEDIUM
3 marks
30 April 2023
29
Which of the following options is/are true?
Multiple correct
MEDIUM
3 marks
11 December 2022
30
If A =
[
a
b
c
d
]
\begin{bmatrix} a & b \\ c & d \end{bmatrix}
[
a
c
b
d
]
is the matrix representation of T with respect to a basis {(1, 1), (1, 2)} for the domain and {(1, 1), (1, -1)} for the co-domain (T defined in the previous comprehension as T(x,y) = (x+y, x-y)), then what is b + c?
Numerical
MEDIUM
2 marks
11 December 2022
31
Let
H
H
H
denote the matrix obtained by putting
k
=
1
k = 1
k
=
1
in the Hessian matrix (obtained in the previous comprehension). Let
T
:
R
3
→
R
3
T: \mathbb{R}^3 \to \mathbb{R}^3
T
:
R
3
→
R
3
be the linear transformation
T
(
x
,
y
,
z
)
=
(
v
1
x
+
v
1
y
+
v
1
z
,
v
2
x
+
v
2
y
+
v
2
z
,
v
3
x
+
v
3
y
+
v
3
z
)
T(x, y, z) = (v_1 x + v_1 y + v_1 z, v_2 x + v_2 y + v_2 z, v_3 x + v_3 y + v_3 z)
T
(
x
,
y
,
z
)
=
(
v
1
x
+
v
1
y
+
v
1
z
,
v
2
x
+
v
2
y
+
v
2
z
,
v
3
x
+
v
3
y
+
v
3
z
)
. If
H
H
H
is the matrix representation of
T
T
T
w.r.t. the standard ordered basis
{
v
1
,
v
2
,
v
3
}
\{v_1, v_2, v_3\}
{
v
1
,
v
2
,
v
3
}
for both domain and codomain, then find the value of
2
(
v
1
+
v
2
)
+
(
v
2
+
v
3
)
−
(
v
1
+
v
3
)
2(v_1 + v_2) + (v_2 + v_3) - (v_1 + v_3)
2
(
v
1
+
v
2
)
+
(
v
2
+
v
3
)
−
(
v
1
+
v
3
)
.
Numerical
MEDIUM
3 marks
11 December 2022
32
Choose the correct option(s) about T.
Comprehension
EASY
2 marks
11 December 2022
33
Let
T
:
R
3
→
R
T: \mathbb{R}^3 \to \mathbb{R}
T
:
R
3
→
R
be a linear transformation such that
T
(
x
,
y
,
z
)
=
det
(
A
)
T(x, y, z) = \det(A)
T
(
x
,
y
,
z
)
=
det
(
A
)
. Then which of following options is/are true?
Comprehension
MEDIUM
2 marks
07 August 2022
34
Let
T
:
R
3
→
R
T: \mathbb{R}^3 \to \mathbb{R}
T
:
R
3
→
R
be a linear transformation such that T(x, y, z) =
det
(
A
)
\text{det}(A)
det
(
A
)
. Let B be the matrix representation of T with respect to some ordered bases for the domain and the codomain. Let m x n be the order of the matrix B and r be the nullity of B. Then find the value of m + n + r.
Comprehension
MEDIUM
2 marks
07 August 2022
35
Find the nullity of T.
Comprehension
EASY
2 marks
07 August 2022
36
Find the rank of T.
Comprehension
EASY
1 marks
07 August 2022
37
Choose the correct option(s).
Comprehension
MEDIUM
2 marks
07 August 2022
38
Let A be the matrix representation of P_W with respect to some orthonormal bases
β
\beta
β
and
γ
\gamma
γ
for V and W, respectively. Then find the dimension of the null space A^T A.
Comprehension
MEDIUM
1 marks
07 August 2022
Showing 38 questions.
Complete question index for this section:
Question 1 (single):
Question 2 (single):
Question 3 (comprehension):
Question 4 (comprehension):
Question 5 (comprehension):
Question 6 (comprehension):
Question 7 (multiple):
Question 8 (comprehension):
Question 9 (comprehension):
Question 10 (comprehension):
Question 11 (comprehension):
Question 12 (comprehension):
Question 13 (comprehension):
Question 14 (comprehension):
Question 15 (comprehension):
Question 16 (comprehension):
Question 17 (comprehension):
Question 18 (comprehension):
Question 19 (comprehension):
Question 20 (comprehension):
Question 21 (comprehension):
Question 22 (comprehension):
Question 23 (comprehension):
Question 24 (comprehension):
Question 25 (comprehension):
Question 26 (comprehension):
Question 27 (comprehension):
Question 28 (comprehension):
Question 29 (multiple):
Question 30 (integer):
Question 31 (integer):
Question 32 (comprehension):
Question 33 (comprehension):
Question 34 (comprehension):
Question 35 (comprehension):
Question 36 (comprehension):
Question 37 (comprehension):
Question 38 (comprehension):
