Week 8 Mathematics 2 Questions

Q1

Find the dimension of

W^\perp

.

Comprehensioneasy3 marksAPRIL 06, 2026

APRIL 06, 2026

Q2

Which of the following sets form an orthonormal basis for

W

? Choose all correct options.

Comprehensioneasy3 marksAPRIL 06, 2026

APRIL 06, 2026

Q3

Suppose

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

represents the projection transformation onto

W

. If

P_W(1,1,2)=(a,b,c)

, find

a+b+c

.

Integer answereasy3 marksAPRIL 06, 2026

APRIL 06, 2026

Q4

Consider the given table of sets of vectors and their properties. Choose the correct option which matches each set of vectors with the appropriate property.

Single correctmedium4 marksAUGUST 03, 2025

AUGUST 03, 2025

Q5

Consider

W=\{(x,y)\in\mathbb{R}^2:x+y=0\}

as a subspace of

\mathbb{R}^2

with the usual inner product. Let

(\alpha,\beta)

be the projection of the vector

(4,-2)

on

W

. Find

\alpha-\beta

.

Integer answermedium3 marksAUGUST 03, 2025

AUGUST 03, 2025

Q6

Consider

\mathbb{R}^2

with the usual inner product. Let

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

be an orthogonal transformation given by

T(x,y)=(ax+by,\ cx+dy)

for all

x,y\in\mathbb{R}

, and let

A

be the matrix representation of

T

with respect to the standard ordered basis. If

a=1

and

\det(A)<0

, find

b+c+d

.

Integer answermedium5 marksAUGUST 03, 2025

AUGUST 03, 2025

Q7

Let

A=\begin{bmatrix}\frac{2}{7}&\frac{6}{7}&\frac{3}{7}\\\frac{3}{7}&\frac{2}{7}&-\frac{6}{7}\\\frac{6}{7}&-\frac{3}{7}&\frac{2}{7}\end{bmatrix}.

Choose the correct option for

A^{-1}

.

Single correctmedium2 marksMARCH 16, 2025

MARCH 16, 2025

Q8

Find

\|T(4,0,-3)\|

.

Comprehensioneasy2 marksMARCH 16, 2025

MARCH 16, 2025

Q9

Suppose

Q

is the matrix representation of

T

with respect to some ordered basis. Find

\det(Q^TQ)

.

Comprehensioneasy2 marksMARCH 16, 2025

MARCH 16, 2025

Q10

Find the angle between

T(1,1,1)

and

T(1,-2,1)

.

Comprehensioneasy2 marksMARCH 16, 2025

MARCH 16, 2025

Q11

If

(a,b,c)

is the projection of

u_2

on

u_1

, find

a+b+c

.

Comprehensioneasy2 marksMARCH 16, 2025

MARCH 16, 2025

Q12

Let

\{v_1,v_2\}

be the orthonormal set of vectors obtained from

\{u_1,u_2\}

by applying the Gram-Schmidt process. Choose the correct option.

Comprehensionmedium2 marksMARCH 16, 2025

MARCH 16, 2025

Q13

Let

(a,b,c)

be the vector obtained by projecting the vector

(1,2,3)

on

W

. Find

a+b+c

.

Comprehensionmedium2 marksMARCH 16, 2025

MARCH 16, 2025

Q14

Find the number of values of

k

for which

u_3

is a unit vector.

Comprehensioneasy1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q15

Find the value of

k

for which

\{u_1,u_2,u_3\}

is an orthonormal basis of

\mathbb{R}^3

.

Comprehensionmedium1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q16

Let

W=\operatorname{span}\{w_1,w_2\}

where

w_1=(1,1,0)

and

w_2=(0,0,1)

. Let

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

map every vector to its projection onto

W

with respect to the standard inner product. Choose all correct statements.

Multiple correctmedium3 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q17

Find

a

.

Comprehensioneasy0.5 marksAUGUST 04, 2024

AUGUST 04, 2024

Q18

Find

b

.

Comprehensioneasy0.5 marksAUGUST 04, 2024

AUGUST 04, 2024

Q19

Find

c

.

Comprehensioneasy0.5 marksAUGUST 04, 2024

AUGUST 04, 2024

Q20

If

\dim(W)=p

and

\dim(W^\perp)=q

, find

p^2+q^2

.

Comprehensioneasy0.5 marksAUGUST 04, 2024

AUGUST 04, 2024

Q21

Find

a

.

Comprehensioneasy0.5 marksAUGUST 04, 2024

AUGUST 04, 2024

Q22

Find

b

.

Comprehensionmedium0.5 marksAUGUST 04, 2024

AUGUST 04, 2024

Q23

Find

c

.

Comprehensionmedium0.5 marksAUGUST 04, 2024

AUGUST 04, 2024

Q24

If

A^T A + A A^T = kI

, where

k

is a scalar and

I

is the identity matrix, find

k

.

Comprehensioneasy0.5 marksAUGUST 04, 2024

AUGUST 04, 2024

Q25

What is

\operatorname{rank}(R_W)

?

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q26

What is the nullity of

R_W

?

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q27

What is

\dim(W^\perp)

?

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q28

Choose the correct options from the following.

Multiple correctmedium4 marksMARCH 24, 2024

MARCH 24, 2024

Q29

Let

A

be an

n\times n

orthogonal matrix. Which statements are true?

Multiple correctmedium3 marksDECEMBER 03, 2023

DECEMBER 03, 2023

Q30

If

\gamma

is the orthonormal basis of

W

obtained from

\beta

by Gram-Schmidt and

(a,b,c)

is the projection of

(1,3,1)

onto

W

, find

a+b+c

.

Comprehensionmedium3 marksDECEMBER 03, 2023

DECEMBER 03, 2023

Q31

Let

A

be a

3\times3

rotation matrix. Choose the correct options.

Multiple correcteasy2 marksAUGUST 06, 2023

AUGUST 06, 2023

Q32

Let

P_W:\mathbb{R}^4\to W

denote the projection map. What is the nullity of

P_W

?

Comprehensioneasy1 marksAUGUST 06, 2023

AUGUST 06, 2023

Q33

If

P_W(0,1,0,1)=(a,b,c,d)

, what is

3(a+b+c+d)

?

Comprehensionhard3 marksAUGUST 06, 2023

AUGUST 06, 2023

Q34

Let

A

be an

n\times n

orthogonal matrix. Choose the correct options.

Multiple correcteasy2 marksAPRIL 02, 2023

APRIL 02, 2023

Q35

If

\{w_1/\|w_1\|,w_2/\|w_2\|\}

denotes the orthonormal basis of

W

obtained by applying Gram-Schmidt on

\beta

, what is

2\|w_2\|^2

?

Comprehensionmedium2 marksAPRIL 02, 2023

APRIL 02, 2023

Q36

Let

P_W

denote the projection of

\mathbb{R}^3

onto

W

. If

P_W(1,0,1)=(a,b,c)

, what is

a+b+c

?

Comprehensionmedium2 marksAPRIL 02, 2023

APRIL 02, 2023

Q37

If

v\in V

has norm

5

, what is the maximum possible norm of

P_W(v)

?

Comprehensioneasy1 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q38

Which of the following options are true?

Comprehensionmedium2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q39

Choose the correct options.

Multiple correctmedium2 marksJULY 10, 2022

JULY 10, 2022

Q40

Let

\beta=\{v_1,v_2\}

be the orthonormal basis of the row space obtained by Gram-Schmidt from the first row and second row of

A

. Let

u_1

be the first orthogonal vector. What is

\|u_1\|^2

?

Comprehensionmedium1 marksJULY 10, 2022

JULY 10, 2022

Q41

For the same Gram-Schmidt process, if

v_2=\frac{1}{\sqrt{195}}(b,c,d)

, what is

b

?

Comprehensionhard1 marksJULY 10, 2022

JULY 10, 2022

Q42

For the same Gram-Schmidt process, if

v_2=\frac{1}{\sqrt{195}}(b,c,d)

, what is

c

?

Comprehensionhard1 marksJULY 10, 2022

JULY 10, 2022

Q43

For the same Gram-Schmidt process, if

v_2=\frac{1}{\sqrt{195}}(b,c,d)

, what is

d

?

Comprehensionhard1 marksJULY 10, 2022

JULY 10, 2022

Mathematics 2 - Week 8