Week 7 Mathematics 2 Questions

Q1

Consider the system of linear equations

Ax=b

given by

-x+y-z=1

,

x-y+z=-1

, and

x+z=0

. Suppose

L

represents the affine space of solutions of

Ax=b

, and let

W

be the subspace in

\mathbb{R}^3

corresponding to the affine space

L

. Choose the correct option.

Single correctmedium4 marksAPRIL 06, 2026

APRIL 06, 2026

Q2

Let

\langle\cdot,\cdot\rangle

denote the standard inner product on

\mathbb{R}^2

, i.e.,

\langle(x_1,x_2),(y_1,y_2)\rangle=x_1y_1+x_2y_2

. For

v\in\mathbb{R}^2

, consider a linear transformation

T_v:\mathbb{R}^2\to\mathbb{R}

defined as

T_v(u)=\langle u,v\rangle

. Which options are true for

T_v

?

Multiple correctmedium4 marksAPRIL 06, 2026

APRIL 06, 2026

Q3

Let

A

and

B

be square matrices of the same order

n

. Which statements are sufficient to conclude that

A

is equivalent to

B

?

Multiple correctmedium4 marksAPRIL 06, 2026

APRIL 06, 2026

Q4

Suppose that

T:\mathbb{R}^3\to\mathbb{R}^2

is a linear transformation given by

T(x,y,z)=(x-2y,\ y+z)

. Let

A

be the matrix representation of

T

with respect to the ordered standard bases for both domain and codomain. If

B

is equivalent to

A

, find the rank of

B

.

Integer answereasy4 marksAPRIL 06, 2026

APRIL 06, 2026

Q5

Consider the set

L=\{(x,y,z)\in\mathbb{R}^3\mid x+y+z=1,\ x-y=0\}

. Find the

y

-coordinate of the point of intersection of the image of

L

under

T

, that is, the set

\{T(x,y,z)\mid (x,y,z)\in L\}

, and the

y

-axis in

\mathbb{R}^2

.

Comprehensionmedium2 marksAPRIL 06, 2026

APRIL 06, 2026

Q6

Let

u=(-1,2,-3)

be a vector from the inner product space

\mathbb{R}^3

with the usual inner product. Which options are true?

Multiple correctmedium6 marksAUGUST 03, 2025

AUGUST 03, 2025

Q7

Let

u=(1,1)

and

v=(v_1,v_2)

be vectors in

\mathbb{R}^2

with the usual inner product. Suppose

\|v\|=2

and the angle between

u

and

v

is

45^\circ

. Find

v_1+v_2

.

Integer answereasy4 marksAUGUST 03, 2025

AUGUST 03, 2025

Q8

Choose all the correct statements.

Comprehensionmedium4 marksAUGUST 03, 2025

AUGUST 03, 2025

Q9

If

\alpha=-2

, find

\beta

.

Comprehensionmedium3 marksAUGUST 03, 2025

AUGUST 03, 2025

Q10

Suppose

T:\mathbb{R}^3\to\mathbb{R}^3

is a linear transformation. Let

M_1

and

M_2

denote the matrix representations of

T

with respect to distinct bases for both domain and codomain

\beta_1

and

\beta_2

, respectively. Choose the correct statements.

Multiple correctmedium4 marksMARCH 16, 2025

MARCH 16, 2025

Q11

Let

A=\begin{bmatrix}-2&0&3\\4&-1&2\end{bmatrix}

. Which of the following matrices are equivalent to

A

?

Multiple correctmedium4 marksMARCH 16, 2025

MARCH 16, 2025

Q12

Let

A=\begin{bmatrix}2&-3\\4&1\end{bmatrix}

. Choose all the correct options.

Multiple correctmedium4 marksMARCH 16, 2025

MARCH 16, 2025

Q13

Let

A=\begin{bmatrix}-1&1\\1&5\end{bmatrix}

and let

B=(b_{ij})

be a matrix similar to

A

. If

b_{11}=7

, find

b_{22}

.

Integer answereasy2 marksMARCH 16, 2025

MARCH 16, 2025

Q14

Which of the affine spaces below correspond to the subspace

W

?

Comprehensionmedium2 marksMARCH 16, 2025

MARCH 16, 2025

Q15

If

A

is equivalent to

I_2

, then find the rank of

A

.

Comprehensioneasy1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q16

Find the value of

k

for which

A

is not equivalent to

I_2

.

Comprehensionmedium1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q17

Find the number of values of

k

for which

A

is similar to

I_2

.

Comprehensionmedium1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q18

For

k=0

, find the angle in degrees between

u_2

and

u_3

.

Comprehensioneasy1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q19

Consider the system

Ax=b

, where

A

is an

m\times n

matrix and

b\in\mathbb{R}^m

. Choose all options that guarantee that the solution set is an affine subspace of

\mathbb{R}^n

.

Multiple correctmedium3 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q20

Consider

P=\frac15\begin{bmatrix}3&4\\-4&3\end{bmatrix}

and

Q=\frac15\begin{bmatrix}3&-4\\4&3\end{bmatrix}

. Let

A

be any

2\times2

matrix and

B=PAQ

. Which statement is true?

Single correctmedium2 marksAUGUST 04, 2024

AUGUST 04, 2024

Q21

If

A

and

B

are similar matrices, then

A^T

and

B^T

are similar matrices.

Comprehensionmedium1 marksAUGUST 04, 2024

AUGUST 04, 2024

Q22

If

A

and

B

have the same rank, then they are similar.

Comprehensioneasy1 marksAUGUST 04, 2024

AUGUST 04, 2024

Q23

Select all true statements.

Comprehensioneasy2 marksAUGUST 04, 2024

AUGUST 04, 2024

Q24

Select all true statements.

Comprehensionmedium2 marksAUGUST 04, 2024

AUGUST 04, 2024

Q25

Let

A

and

B

be

n\times n

similar matrices. Suppose

A

has exactly

n-1

linearly independent columns. Then

\det(B)

is equal to _____.

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q26

Let

A

and

B

be

n\times n

matrices. Which statements are true?

Multiple correctmedium3 marksDECEMBER 03, 2023

DECEMBER 03, 2023

Q27

Choose the correct options from the following.

Comprehensionmedium2 marksDECEMBER 03, 2023

DECEMBER 03, 2023

Q28

If the dimension of

L

is

m

and the dimension of

L'

is

n

, then find

m+n

.

Comprehensioneasy1 marksDECEMBER 03, 2023

DECEMBER 03, 2023

Q29

An inner product on a vector space

V

satisfies positive-definiteness, additivity in the first input, symmetry, and scalar homogeneity. Let

V=\mathbb{R}^2

and define

\langle (x_1,x_2),(y_1,y_2)\rangle=x_1y_1-x_2y_1-x_2y_2

. Which conditions are satisfied?

Multiple correctmedium3 marksAUGUST 06, 2023

AUGUST 06, 2023

Q30

Let

B

denote the matrix of

T

with respect to the standard ordered basis for both domain and codomain. Choose the correct options.

Comprehensionmedium2 marksAUGUST 06, 2023

AUGUST 06, 2023

Q31

An inner product on

V

satisfies positive-definiteness, additivity, symmetry, and homogeneity. Let

V=\mathbb{R}^2

and define

\langle(x_1,x_2),(y_1,y_2)\rangle=x_1y_1-x_1y_2+x_2y_2

. Which conditions are satisfied?

Multiple correctmedium3 marksAPRIL 02, 2023

APRIL 02, 2023

Q32

Let

B

denote the matrix of

T

with respect to the standard ordered bases for

\mathbb{R}^2

and

\mathbb{R}^3

. Choose the correct options.

Comprehensionmedium2 marksAPRIL 02, 2023

APRIL 02, 2023

Q33

What is the dimension of

L

?

Comprehensioneasy1 marksAPRIL 02, 2023

APRIL 02, 2023

Q34

Which of the following options are true?

Multiple correctmedium2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q35

Find

\|(1,3)\|^2

.

Comprehensioneasy1 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q36

Which of the following are unit vectors in

V

?

Comprehensionmedium2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q37

Which affine subspace was considered by Soumya?

Comprehensionmedium2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q38

Which affine subspace was considered by Sohini?

Comprehensionmedium2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q39

Which function represents

f

correctly?

Comprehensionhard2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q40

An inner product on

V

satisfies positive-definiteness, additivity, symmetry, and homogeneity. Let

V=\mathbb{R}^2

and define

\langle(x_1,x_2),(y_1,y_2)\rangle=x_1y_1-x_1y_2-x_2y_1+x_2y_2

. Which conditions are satisfied?

Multiple correctmedium2 marksJULY 10, 2022

JULY 10, 2022

Q41

Let

U

be a subspace of

\mathbb{R}^3

with basis

\{(1,0,1),(0,1,2)\}

. Which subsets of

\mathbb{R}^3

are appropriate candidates for affine subspaces whose corresponding vector subspace is

U

?

Multiple correctmedium2 marksJULY 10, 2022

JULY 10, 2022

Q42

Let

A=\begin{bmatrix}1&0\\1&1\end{bmatrix}

,

B=\begin{bmatrix}1&1\\0&1\end{bmatrix}

, and

C=\begin{bmatrix}1&0\\0&1\end{bmatrix}

. For Pair I:

A,B

; Pair II:

A,C

; Pair III:

B,C

, choose the correct option.

Single correctmedium2 marksJULY 10, 2022

JULY 10, 2022

Q43

A norm on

V

satisfies triangle inequality, scalar homogeneity, and positive-definiteness. Consider

\| (x_1,x_2,x_3)\|=|x_1+x_2+x_3|

on

\mathbb{R}^3

. Which conditions are satisfied?

Multiple correctmedium1 marksJULY 10, 2022

JULY 10, 2022

Mathematics 2 - Week 7