Recent questions and answers in Linear Algebra / GATE Overflow for GATE CSE

6 votes

2 answers

1

GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 1
Let $A$ be an $n \times n$ matrix of real or complex numbers. Which of the following statements are equivalent to: “the matrix $A$ is invertible”? The columns of $A$ are linearly independent. The rows of $A$ are linearly independent. The only solution of the homogeneous equations $Ax = 0$ is $x = 0$. The rank of $A$ is $n$.

Let $A$ be an $n \times n$ matrix of real or complex numbers. Which of the following statements are equivalent to: “the matrix $A$ is invertible”?The columns of $A$ a...

Teet Makor

124 views

Teet Makor answered 5 days ago

16 votes

5 answers

2

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 11
Which of the following is(are) true for the following system of linear equations $\text{AX}=\overrightarrow{0}$ ... $\text{AX}=\overrightarrow{0}$.

Which of the following is(are) true for the following system of linear equations $\text{AX}=\overrightarrow{0}$$$\left[\begin{array}{ccc}2 & 3 & -5 \\-5 & -1 & 32 \\2 & -...

Biswajit Kumar

1.2k views

Biswajit Kumar answered 6 days ago

2 votes

1 answer

3

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 9
Suppose that $a \neq 0$ and $a \neq b$. Which equation below is the equation relating $a, b$ and $c$ ... $4 a-3 b+c \neq 0$ $3 a-4 b-c \neq 0$ $4 a-3 b-c \neq 0$

Suppose that $a \neq 0$ and $a \neq b$. Which equation below is the equation relating $a, b$ and $c$ so that the vectors$$\left[\begin{array}{l}1 \\1 \\a\end{array}\right...

tejashmore25

50 views

tejashmore25 answered Apr 18

1 votes

1 answer

4

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 8
Consider a linear system $A \mathbf{x}=\mathbf{b}$, where $A$ is a $3 \times 4$ matrix with $\operatorname{Rank}(A)=2$ ... unique solution. (i) infinitely many solutions, (ii) no solution. (i) infinitely many solutions, (ii) unique solution.

Consider a linear system $A \mathbf{x}=\mathbf{b}$, where $A$ is a $3 \times 4$ matrix with $\operatorname{Rank}(A)=2$.How many solutions does this system have if $(i) \o...

tejashmore25

47 views

tejashmore25 answered Apr 18

1 votes

1 answer

5

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 10
Let $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ ... $\left(\begin{array}{lll}2 & 1 & 3 \\ 6 & 3 & 9\end{array}\right)$

Let $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ be a linear transformation such that $T\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right]=\left[\begin{array}{l}2 \\ 6...

tejashmore25

49 views

tejashmore25 answered Apr 18

4 votes

1 answer

6

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 7
Consider a $4 \times 4$ matrix $A$ and a vector $\mathbf{v} \in \mathbb{R}^{4}$ such that $A^{4} \mathbf{v}=\mathbf{0}$ but $A^{3} \mathbf{v} \neq \mathbf{0}$ ... $\mathbb{R}^{4}$. $\mathcal{B}$ is a basis of $\mathbb{R}^{4}$. $\mathcal{B}$ is not linearly independent.

Consider a $4 \times 4$ matrix $A$ and a vector $\mathbf{v} \in \mathbb{R}^{4}$ such that $A^{4} \mathbf{v}=\mathbf{0}$ but $A^{3} \mathbf{v} \neq \mathbf{0}$.Set $\mathc...

tejashmore25

58 views

tejashmore25 answered Apr 18

5 votes

2 answers

7

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 5
Consider the linear map $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined by $ T(x, y)=(x-y, x-2 y), \text { for } x, y \in \mathbb{R} $ Let $\mathcal{E}$ be the standard basis for $\mathbb{R}^{2}$ ... $\left(\begin{array}{ll}0 & -1 \\ 1 & -1\end{array}\right)$

Consider the linear map $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined by$$T(x, y)=(x-y, x-2 y), \text { for } x, y \in \mathbb{R}$$Let $\mathcal{E}$ be the stand...

tejashmore25

62 views

tejashmore25 answered Apr 18

2 votes

1 answer

8

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 14
Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$ ... the matrix equation $A x=0$ has infinitely many solutions, then $\operatorname{rank}(A) \leq 4$.

Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$. Which of the following statements are true?$\op...

harshit8118

56 views

harshit8118 answered Apr 18

5 votes

2 answers

9

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 13
Suppose $A$ is a $4 \times 3$ matrix and $B$ is a $3 \times 2$ matrix, and let $T$ be the matrix transformation $T(x)=A B x$. Which of the following must be true? The column space of $A B$ ... space of $A$. $T$ has domain $\mathbf{R}^{2}$ and codomain $\mathbf{R}^{4}$. $T$ cannot be onto.

Suppose $A$ is a $4 \times 3$ matrix and $B$ is a $3 \times 2$ matrix, and let $T$ be the matrix transformation $T(x)=A B x$. Which of the following must be true?The colu...

tejashmore25

55 views

tejashmore25 answered Apr 18

3 votes

2 answers

10

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 12
Consider two statements S1 and S2. S1: If $\left\{v_{1}, \ldots, v_{n}\right\}$ are linearly INDEPENDENT vectors in $V$, then $\left\{T\left(v_{1}\right), \ldots, T\left(v_{n}\right)\right\}$ are linearly ... $\mathrm{S} 2$ is true. Both S1 and S2 are true. Both S1 and S2 are false.

Consider two statements S1 and S2.S1: If $\left\{v_{1}, \ldots, v_{n}\right\}$ are linearly INDEPENDENT vectors in $V$, then $\left\{T\left(v_{1}\right), \ldots, T\left(v...

tejashmore25

64 views

tejashmore25 answered Apr 18

1 votes

1 answer

11

ME: GATE-2005
A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. The highest possible rank of A is:____________

A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. Thehighest possible rank of A is:____________

therealvs

4.3k views

therealvs answered Apr 16

3 votes

1 answer

12

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 11
Which of the following statements are true? There exists a $3 \times 3$ matrix $A$ and vectors $b, c \in \mathbb{R}^{3}$ such that the linear system $A x=b$ has a unique solution but $A x=c$ has infinitely ... $n$, then the column space of $A$ is equal to the column space of $B$.

Which of the following statements are true?There exists a $3 \times 3$ matrix $A$ and vectors $b, c \in \mathbb{R}^{3}$ such that the linear system $A x=b$ has a unique s...

Tamilhp

42 views

Tamilhp answered Apr 14

3 votes

1 answer

13

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 3
Let the linear transformation $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ be defined by $T\left(x_{1}, x_{2}\right)=\left(x_{1}, x_{1}+x_{2}, x_{2}\right)$. Then the nullity of $T$ is: 0 1 2 3

Let the linear transformation $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ be defined by $T\left(x_{1}, x_{2}\right)=\left(x_{1}, x_{1}+x_{2}, x_{2}\right)$. Then the n...

Dhananjaykumar4u

56 views

Dhananjaykumar4u answered Apr 12

4 votes

1 answer

14

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 16
Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$ ... the matrix equation $A x=0$ has infinitely many solutions, then $\operatorname{rank}(A) \leq 4$.

Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$. Which of the following statements are true?$\op...

Ajay Sreenivas

56 views

Ajay Sreenivas answered Apr 12

4 votes

1 answer

15

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 2
Suppose that $\left\{\mathbf{v}_{\mathbf{1}}, \mathbf{v}_{\mathbf{2}}, \mathbf{v}_{\mathbf{3}}\right\}$ is a linearly independent set of vectors in $\mathbb{R}^{6}$ ... is linearly independent $\left\{\mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{w}\right\}$ is linearly independent

Suppose that $\left\{\mathbf{v}_{\mathbf{1}}, \mathbf{v}_{\mathbf{2}}, \mathbf{v}_{\mathbf{3}}\right\}$ is a linearly independent set of vectors in $\mathbb{R}^{6}$.Furth...

sadhak

76 views

sadhak answered Apr 12

4 votes

1 answer

16

GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 1
Let $T_{1}, T_{2}: R^{5} \rightarrow R^{3}$ be linear transformations s.t $\operatorname{rank}\left(T_{1}\right)=3$ and nullity $\left(T_{2}\right)=3$. Let $T_{3}: R^{3} \rightarrow R^{3}$ be linear transformation s.t $T_{3}\left(T_{1}\right)=T_{2}$. Then find rank of $T_{3}$

Let $T_{1}, T_{2}: R^{5} \rightarrow R^{3}$ be linear transformations s.t $\operatorname{rank}\left(T_{1}\right)=3$ and nullity $\left(T_{2}\right)=3$. Let $T_{3}: R^{3} ...

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