Week 6 Mathematics 2 Questions

Q1

Let

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

be a linear transformation such that

T^2=T

, and

\operatorname{rank}(T)=2

. Which of the following statements is correct?

Single correctmedium4 marksAPRIL 06, 2026

APRIL 06, 2026

Q2

Let

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

be a linear transformation such that

T(1,0,1)=(1,2,1)

,

T(0,1,1)=(0,1,1)

, and

T(1,1,0)=(1,1,0)

. Let

U=\operatorname{span}\{(1,0,1),(0,1,1)\}

. Choose all correct options.

Multiple correctmedium4 marksAPRIL 06, 2026

APRIL 06, 2026

Q3

Let

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

be defined by

T(x,y,z)=(x+y,\ y+z,\ z+x+ky)

, where

k\in\mathbb{R}

. Which statements are true?

Multiple correctmedium4 marksAPRIL 06, 2026

APRIL 06, 2026

Q4

Let

v=(a,b,c)\in\mathbb{R}^3

be a vector in the kernel of

T

such that

a+b+c=1

. Find the value of

a+2b+3c

up to two decimal places.

Comprehensionmedium4 marksAPRIL 06, 2026

APRIL 06, 2026

Q5

Find the dimension of the image of

S

.

Comprehensionmedium4 marksAPRIL 06, 2026

APRIL 06, 2026

Q6

Consider a vector

v\in U

such that

S(v)\in\operatorname{span}\{(1,1,1)\}

, and

\|v\|^2=6

, where

\|\cdot\|

is the norm with respect to the standard inner product on

\mathbb{R}^3

. Which options are the possible choices for

v

?

Comprehensionmedium3 marksAPRIL 06, 2026

APRIL 06, 2026

Q7

Let

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

be the linear transformation determined by

T(1,0)=(\alpha-6,-4)

and

T(0,1)=(3,\alpha+1)

, where

\alpha\in\mathbb{R}

. Let

\alpha_1

and

\alpha_2

be the minimum and maximum values, respectively, of

\alpha

for which

T

fails to be injective. Find

\alpha_2^2-\alpha_1^2

.

Integer answermedium5 marksAUGUST 03, 2025

AUGUST 03, 2025

Q8

Find the number of columns in the matrix, with respect to any bases, of the linear transformation

T

.

Comprehensioneasy2 marksAUGUST 03, 2025

AUGUST 03, 2025

Q9

Find the value of

k

for which the set

\{(1,1,k^2,k-1),(1,1,0,-1)\}

is a basis for the kernel of

T

.

Comprehensionmedium3 marksAUGUST 03, 2025

AUGUST 03, 2025

Q10

Which of the following is the matrix of the linear transformation

T

with respect to standard bases for both the domain and the codomain?

Comprehensioneasy2 marksAUGUST 03, 2025

AUGUST 03, 2025

Q11

Which of the following spaces are isomorphic to the kernel of

T

?

Comprehensionmedium5 marksAUGUST 03, 2025

AUGUST 03, 2025

Q12

Which of the following statements is true?

Comprehensionmedium4 marksMARCH 16, 2025

MARCH 16, 2025

Q13

For the value of

k

obtained in the previous question, which of the following vectors belong to the range of

T

?

Comprehensionmedium4 marksMARCH 16, 2025

MARCH 16, 2025

Q14

Find

a

.

Comprehensioneasy1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q15

Find

b

.

Comprehensioneasy1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q16

Is

T

injective or surjective? If

T

is injective, write

1

and if

T

is surjective, write

-1

.

Comprehensionmedium1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q17

Let

T:\mathbb{R}^n\to\mathbb{R}^m

be a linear transformation. Choose the correct option that guarantees that

T

is bijective.

Single correctmedium3 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q18

Let

S

be any straight line in

\mathbb{R}^2

passing through the origin. If

T(S)

is the image of

S

under

T

in

\mathbb{R}^3

, what is the dimension of

T(S)

?

Comprehensionmedium1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q19

Let

B_1=\{(1,-1),(0,2)\}

and

B_2=\{(1,0,0),(0,1,0),(0,0,1)\}

. Which matrix is the representation of

T

with respect to

B_1

for the domain and

B_2

for the codomain?

Comprehensioneasy1 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q20

Let

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

. Let

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

be defined by

T(x,y,z)=(cx-y,y+z)

. Choose all correct options.

Multiple correcthard3 marksDECEMBER 01, 2024

DECEMBER 01, 2024

Q21

Let

A=\begin{bmatrix}x&-4\\3&y\end{bmatrix}

and

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

be the linear transformation given by

T(x,y)=\operatorname{trace}(A)

. Which statement is true?

Single correcteasy2 marksAUGUST 04, 2024

AUGUST 04, 2024

Q22

Which matrix represents

T

with respect to the standard ordered basis for both the domain and codomain?

Comprehensionmedium2 marksAUGUST 04, 2024

AUGUST 04, 2024

Q23

Which option defines the transformation

T

?

Comprehensioneasy1 marksAUGUST 04, 2024

AUGUST 04, 2024

Q24

Find the value of

a

.

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q25

Find the value of

b

.

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q26

Find the value of

c

.

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q27

Find the value of

d

.

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q28

Choose the correct option from the following.

Comprehensionmedium2 marksMARCH 24, 2024

MARCH 24, 2024

Q29

Choose the correct options from the following.

Comprehensionmedium2 marksMARCH 24, 2024

MARCH 24, 2024

Q30

Which of the following statements is true?

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q31

Is the statement true or false? If

\{v_1,v_2,v_3\}

are linearly independent vectors in

V

, then

\{T(v_1),T(v_2),T(v_3)\}

are linearly independent vectors in

W

.

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q32

Is the statement true or false? Let

\{u_1,u_2,u_3\}

be a subset of

V

. If

\{T(u_1),T(u_2),T(u_3)\}

is linearly independent in

W

, then

\{u_1,u_2,u_3\}

could also be linearly dependent in

V

.

Comprehensionmedium1 marksMARCH 24, 2024

MARCH 24, 2024

Q33

Find the nullity of

T

.

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q34

Which of the following is a basis for the kernel of

T

?

Comprehensioneasy1 marksMARCH 24, 2024

MARCH 24, 2024

Q35

Which of the following is a basis for the image of

T

?

Comprehensionmedium1 marksMARCH 24, 2024

MARCH 24, 2024

Q36

Let

T

be a linear transformation from

\mathbb{R}^2

to

\mathbb{R}^3

. Let

S\subset\mathbb{R}^2

be a line passing through the origin. Which of the following are possible?

Multiple correcteasy2 marksDECEMBER 03, 2023

DECEMBER 03, 2023

Q37

Find the matrix corresponding to

T

if

\gamma

is used as the basis for both the domain and codomain.

Comprehensionmedium3 marksDECEMBER 03, 2023

DECEMBER 03, 2023

Q38

Is

T

an isomorphism?

Comprehensioneasy1 marksDECEMBER 03, 2023

DECEMBER 03, 2023

Q39

Which of the following options are true?

Multiple correctmedium2 marksAUGUST 06, 2023

AUGUST 06, 2023

Q40

Consider

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

and

T:V\to\mathbb{R}^4

defined by

T(A)=(a,b,c,a+b-c)

. Choose the correct options.

Multiple correctmedium2 marksAUGUST 06, 2023

AUGUST 06, 2023

Q41

Define a linear transformation

T:V_2\to\mathbb{R}^2

by

T(x,y,z)=(x,x+y+z)

. What is the rank of

T

?

Integer answermedium2 marksAUGUST 06, 2023

AUGUST 06, 2023

Q42

Let

S:V_1\to V_2

be a linear transformation. If

m\times n

is the order of the matrix

D

of

S

with respect to ordered bases for

V_1

and

V_2

, what is

2m-3n

?

Integer answereasy1 marksAUGUST 06, 2023

AUGUST 06, 2023

Q43

If

A=\begin{bmatrix}a&b&c\d&e&f\g&h&i\end{bmatrix}

denotes the matrix of

T

with respect to

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

for domain and codomain, what is

2b+2e+2h

?

Comprehensionhard2 marksAUGUST 06, 2023

AUGUST 06, 2023

Q44

Consider the vector space

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

and

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

defined by

T(x,y,z)=\begin{bmatrix}x+y&x+y+z\\x+y&x+y+z\end{bmatrix}

. Choose the correct option.

Single correctmedium2 marksAPRIL 02, 2023

APRIL 02, 2023

Q45

Let

V=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\in M_{2\times2}(\mathbb{R}):a+b=c+d\right\}

and

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

be onto. What is the dimension of

\ker(T)

?

Integer answereasy3 marksAPRIL 02, 2023

APRIL 02, 2023

Q46

If

A=\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}

denotes the matrix of

T

with respect to

\{(1,1),(1,-1)\}

for

\mathbb{R}^2

and

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

for

\mathbb{R}^3

, what is

a+d+e

?

Comprehensionmedium3 marksAPRIL 02, 2023

APRIL 02, 2023

Q47

Define

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

by

T(x,y,z)=(0,x-z)

. What is the rank of

T

?

Comprehensioneasy1 marksAPRIL 02, 2023

APRIL 02, 2023

Q48

If the

m\times n

matrix

B

is the matrix of

T

with respect to some basis for

W

and the standard ordered basis for

\mathbb{R}^2

, what is

m+n

?

Comprehensioneasy1 marksAPRIL 02, 2023

APRIL 02, 2023

Q49

If

A

is the matrix representation of

S\circ T

with respect to

\beta

for both domain and codomain, and the order of

A

is

m\times n

, find

m+n

.

Comprehensioneasy1 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q50

If

K=\{(x,y,z): ax+by=0,\ cy+dz=0\}

is the null space of

S\circ T

, find

\frac{a}{b}-2\frac{c}{d}

.

Comprehensionmedium2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q51

Find the rank of

S\circ T

.

Comprehensioneasy1 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q52

Find the nullity of the matrix

A

.

Comprehensioneasy1 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q53

Which of the following options is true?

Comprehensionmedium2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q54

If

T(x,y,z)=(mx+ny+az,px+qy+rz)

, find

(m+n+a)-3(p+q+r)

.

Comprehensionhard2 marksNOVEMBER 20, 2022

NOVEMBER 20, 2022

Q55

What is the value of

d-a

?

Comprehensionmedium1 marksJULY 10, 2022

JULY 10, 2022

Q56

What is the value of

e-b

?

Comprehensionmedium1 marksJULY 10, 2022

JULY 10, 2022

Q57

What is the value of

f-c

?

Comprehensionmedium1 marksJULY 10, 2022

JULY 10, 2022

Q58

What is the value of

m

?

Comprehensioneasy1 marksJULY 10, 2022

JULY 10, 2022

Q59

What is the value of

n

?

Comprehensioneasy1 marksJULY 10, 2022

JULY 10, 2022

Q60

If

A

is the matrix representation of

T

with respect to the standard bases of

\mathbb{R}^3

and

\mathbb{R}^2

, then

A

is:

Comprehensioneasy1 marksJULY 10, 2022

JULY 10, 2022

Q61

If

\{(l,m,n)\}

is a basis of

\ker(T)

, find

l

when

n=1

.

Comprehensionmedium1 marksJULY 10, 2022

JULY 10, 2022

Q62

If

\{(l,m,n)\}

is a basis of

\ker(T)

, find

m

when

n=1

.

Comprehensionmedium1 marksJULY 10, 2022

JULY 10, 2022

Mathematics 2 - Week 6