Week 4 Mathematics 2 Questions

Q1

Consider a subspace

W=\{(x,y,z,w)\in\mathbb{R}^4\mid x+y+z=0,\;y+z=w\}

of

\mathbb{R}^4

. Which of the following options describes a basis of

W

?

Single correctMARCH 15, 2026

MARCH 15, 2026

Q2

Find

k_1

.

ComprehensionMARCH 15, 2026

MARCH 15, 2026

Q3

Find

k_2

.

ComprehensionMARCH 15, 2026

MARCH 15, 2026

Q4

Consider the vector space

(W,+,\cdot)

with the addition and scalar multiplication operations from the correct option of the previous question. What is the dimension of the vector space?

ComprehensionMARCH 15, 2026

MARCH 15, 2026

Q5

The trace of a square matrix

A

is defined as the sum of its diagonal entries. Consider the set

W

whose elements are

3\times 3

matrices with zero trace, i.e.

W=\left\{\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\in M_{3\times3}(\mathbb{R})\mid a_{11}+a_{22}+a_{33}=0\right\}.

W

is a vector space with the standard operations of addition and scalar multiplication. Find the dimension of

W

.

Integer answerOCTOBER 26, 2025

OCTOBER 26, 2025

Q6

Let

A

be a

5\times7

matrix. If

n_1

and

n_2

are the minimum and maximum possible values for the rank of

A

, respectively, then find the value of

n_2-n_1

.

Integer answerOCTOBER 26, 2025

OCTOBER 26, 2025

Q7

Consider the matrix

\begin{bmatrix}2&-3&1&-4\\3&1&-4&5\\4&k&-6&8\end{bmatrix}.

Find the value of

k

such that the rank of the given matrix is

2

.

Integer answerOCTOBER 26, 2025

OCTOBER 26, 2025

Q8

For

a=0

, find the value of

b

such that the set

\{v_1,v_2,v_3\}

is not a basis of

\mathbb{R}^3

.

ComprehensionOCTOBER 26, 2025

OCTOBER 26, 2025

Q9

Which of the following is a basis for

V=\{A\in M_{2\times2}(\mathbb{R})\mid A^T=A\}

, the vector space of

2\times2

real symmetric matrices?

Single correctJULY 13, 2025

JULY 13, 2025

Q10

Given a vector space

V

, which of the following statements are equivalent to the statement that a set

B\subset V

is a basis?

Multiple correctJULY 13, 2025

JULY 13, 2025

Q11

Find the rank of

A

.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q12

Find the maximum possible rank of the matrix

A

.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q13

Find the value of

k

for which rank of the matrix

A

is equal to

2

.

ComprehensionJULY 13, 2025

JULY 13, 2025

Q14

Let

v_1,v_2,v_3

be linearly independent vectors in

\mathbb{R}^3

. Let

A\in M_{3\times3}(\mathbb{R})

be a matrix such that

v_1-v_2

,

v_2-v_3

,

v_1+v_2

are the columns of

A

. Let

B\in M_{3\times3}(\mathbb{R})

be a matrix such that

v_1+v_2+v_3

,

2v_1+3v_2

,

2v_3-v_2

are the columns of

B

. Which of the following options is correct?

Single correctFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q15

The trace of a square matrix

A

is defined as the sum of its diagonal entries. Consider the set

W

whose elements are

3\times3

matrices with zero trace, i.e.,

W=\left\{\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\in M_{3\times3}(\mathbb{R})\mid a_{11}+a_{22}+a_{33}=0\right\}.

W

is a vector space with the standard operations of addition and scalar multiplication. Find the dimension of

W

.

Integer answerFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q16

Let

A

be a

5\times7

matrix. If

n_1

and

n_2

are the minimum and maximum possible values for the rank of

A

, respectively, then find the value of

n_2-n_1

.

Integer answerFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q17

Consider the matrix

\begin{bmatrix}2&-3&1&-4\\3&1&-4&5\\4&k&-6&8\end{bmatrix}.

Find the value of

k

such that the rank of the given matrix is

2

.

Integer answerFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q18

For

a=0

, find the value of

b

such that the set

\{v_1,v_2,v_3\}

is not a basis of

\mathbb{R}^3

.

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q19

Choose the correct set of matrices that forms a basis for

W_1

.

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q20

Choose all the correct sets of matrices that span

W_2

.

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q21

What is the dimension of

W_1\cap W_2

?

ComprehensionFEBRUARY 23, 2025

FEBRUARY 23, 2025

Q22

Choose all the correct options from the following.

Multiple correctOCTOBER 27, 2024

OCTOBER 27, 2024

Q23

W_2=\operatorname{span}\{(1,-1,1),(4,1,2),(2,3,0)\}

.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q24

W_3

is the set of all

2\times2

symmetric matrices.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q25

W_1=\{A\in M_{2\times2}(\mathbb{R}):A^T=-A\}

.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q26

W_2

is the set of all

2\times2

matrices such that the sum of entries in each row is zero.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q27

W_3

is the set of all

2\times2

matrices such that the sum of the diagonal entries is zero.

ComprehensionOCTOBER 27, 2024

OCTOBER 27, 2024

Q28

Consider the following subsets of

\mathbb{R}^4

.

W=\operatorname{span}\{(2,-1,0,4),(-1,1,0,3),(1,2,0,3),(2,2,0,10)\}

,

B_1=\{(1,0,0,0),(0,1,0,0),(0,0,0,1)\}

, and

B_2=\{(2,-1,0,4),(-1,1,0,3),(1,2,0,3)\}

. Select the correct option.

Single correctJULY 07, 2024

JULY 07, 2024

Q29

U=\{(a,b,c,d,e):a+b+c+d+e=0 ext{ and } a,b,c,d,e\in\mathbb{R}\}

.

ComprehensionJULY 07, 2024

JULY 07, 2024

Q30

V=\left\{\begin{bmatrix}a&b&0\\c&0&a+b\end{bmatrix}:a,b,c\in\mathbb{R}\right\}

.

ComprehensionJULY 07, 2024

JULY 07, 2024

Q31

W=\operatorname{span}\{(1,1,1),(1,0,-1),(2,1,0)\}

.

ComprehensionJULY 07, 2024

JULY 07, 2024

Q32

W_2=\left\{\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}:a,b,c\in\mathbb{R}\text{ such that }a=b=c\right\}

. If

W_2

is a subspace, find the dimension; else write the answer as

0

.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q33

W_3=\left\{\begin{bmatrix}a&0&0\\0&b&0\\0&0&c\end{bmatrix}:a,b,c\in\mathbb{R}\right\}

. If

W_3

is a subspace, find the dimension; else write the answer as

0

.

ComprehensionFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q34

Consider the vectors

v_1=(1,-1,0)

,

v_2=(2,3,-1)

, and

v_3=(a,b,c)

in

\mathbb{R}^3

. Choose the correct options from the following.

Multiple correctFEBRUARY 25, 2024

FEBRUARY 25, 2024

Q35

Choose the correct option(s) from the following statements.

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q36

Find

\dim(W_1+W_2)

.

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q37

Let

A\in W

be a non-zero matrix. Then find

\operatorname{rank}(A)

.

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q38

What is the dimension of the vector space

W

?

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q39

Which of the following sets form a basis for

W

?

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q40

What is the rank of

A

?

ComprehensionOCTOBER 29, 2023

OCTOBER 29, 2023

Q41

V_1=\{(x,y,z)\in\mathbb{R}^3:2x+3y=0=2z+3x\}

with usual addition and scalar multiplication. Find

\dim(V_1)

.

ComprehensionJULY 16, 2023

JULY 16, 2023

Q42

V_2=\{A\in M_3(\mathbb{R}):\text{sum of the diagonal entries of }A\text{ is }0\text{ and sum of each row is }0\}

with usual addition and scalar multiplication of matrices. Find

\dim(V_2)

.

ComprehensionJULY 16, 2023

JULY 16, 2023

Q43

What is the rank of

A

?

ComprehensionJULY 16, 2023

JULY 16, 2023

Q44

Subset 1 is a subspace of dimension __________. Enter the numerical value only.

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q45

Subset 2 is a subspace of dimension __________. Enter the numerical value only.

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q46

Subset 3 is a subspace of dimension __________. Enter the numerical value only.

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q47

Subset 4 is a subspace of dimension __________. Enter the numerical value only.

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q48

Which of the following option(s) represent

W_1\cap W_2

?

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q49

NOTE: Enter your answer to the nearest integer. What is the dimension of

W_1\cap W_2

?

ComprehensionOCTOBER 16, 2022

OCTOBER 16, 2022

Q50

If

W=\operatorname{Span}\{(1,1,-1),(3,-2,0),(5,0,-2),(0,5,-3)\}

, then find the dimension of

W

.

Integer answerJUNE 05, 2022

JUNE 05, 2022

Q51

What is the dimension of

W_1\cap W_2

?

ComprehensionJUNE 05, 2022

JUNE 05, 2022

Q52

What is the dimension of

W_1

?

ComprehensionJUNE 05, 2022

JUNE 05, 2022

Q53

S=\{(1,0,1),(0,1,1),(1,1,0)\}

is a ______ of

\mathbb{R}^3

. Enter

3

best possible options as serial numbers in increasing order without commas or spaces.

ComprehensionJUNE 05, 2022

JUNE 05, 2022

Q54

A spanning set of

\mathbb{R}^2

with

2

elements must be a ______. Enter

2

best possible options as serial numbers in increasing order without commas or spaces.

ComprehensionJUNE 05, 2022

JUNE 05, 2022

All questions