Recent questions tagged linear-algebra / GATE Overflow for GATE CSE

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241

#Eigen Vectors
Find the eigen values and eigen vector of the following matrix????

Find the eigen values and eigen vector of the following matrix????

Çșȇ ʛấẗẻ

1.2k views

Çșȇ ʛấẗẻ asked Mar 21, 2023

6 votes

4 answers

242

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 3
Let us consider the following three vectors $v_1, v_2,$ and $v_3$ in $\mathbb{R}^3$ ... $\left\{v_1, v_3\right\}$ are linearly independent.

Let us consider the following three vectors $v_1, v_2,$ and $v_3$ in $\mathbb{R}^3$.$$\begin{aligned}& v_1=\left(\begin{array}{lll}1 & 0 & -2\end{array}\right) \\& v_2=\l...

GO Classes

795 views

GO Classes asked Mar 14, 2023

5 votes

3 answers

243

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 4
Let $a, b$ be in $\mathbb{R}$ ... and $b=1$ $a \neq 0$ and $b \neq 1$ $a=0$ and $b \neq 1$ $a \neq 0$ and $b=1$

Let $a, b$ be in $\mathbb{R}$. Consider the three vectors$$\boldsymbol{v}_1=\left[\begin{array}{l}a \\0 \\0\end{array}\right], \quad \boldsymbol{v}_2=\left[\begin{array}{...

GO Classes

368 views

GO Classes asked Mar 14, 2023

8 votes

3 answers

244

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 5
Which of the following is FALSE? If $v_1, \ldots, v_4$ are in $\mathbf{R}^5$ and $\left\{v_1, v_2, v_3\right\}$ is linearly dependent then $\left\{v_1, v_2, v_3, v_4\right\}$ is linearly dependent. ... $6$ vectors in $\mathbf{R}^5$ is linearly dependent.

Which of the following is FALSE?If $v_1, \ldots, v_4$ are in $\mathbf{R}^5$ and $\left\{v_1, v_2, v_3\right\}$ is linearly dependent then $\left\{v_1, v_2, v_3, v_4\right...

GO Classes

672 views

GO Classes asked Mar 14, 2023

12 votes

2 answers

245

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 6
Consider a set of $n$ linearly independent vectors $\left\{\vec{w}_1, \ldots, \vec{w}_n\right\} \in \mathbb{R}^n$. A vector $\vec{u} \in \mathbb{R}^n$ will: Option $1.$ Always be a linear ... $2$ is correct then enter $\text{ 2 }.$

Consider a set of $n$ linearly independent vectors $\left\{\vec{w}_1, \ldots, \vec{w}_n\right\} \in \mathbb{R}^n$. A vector $\vec{u} \in \mathbb{R}^n$ will:Option $1.$ Al...

GO Classes

759 views

GO Classes asked Mar 14, 2023

9 votes

3 answers

246

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 7
Which of the following set is/are linearly independent? $\left\{\left[\begin{array}{l}1 \\ 2\end{array}\right],\left[\begin{array}{l}2 \\ 1\end{array}\right]\right\}$ ...

Which of the following set is/are linearly independent?$\left\{\left[\begin{array}{l}1 \\ 2\end{array}\right],\left[\begin{array}{l}2 \\ 1\end{array}\right]\right\}$ $\le...

GO Classes

765 views

GO Classes asked Mar 14, 2023

11 votes

7 answers

247

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 8
$a\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4 \\ 5\end{array}\right]+b\left[\begin{array}{c}-1 \\ 2 \\ -3 \\ 4 \\ -5\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0 \\ 0 \\ 0\end{array}\right]$ How many number of pairs $(a, b)$ are there, that satisfy the above equation? $0$ $1$ Infinite $2$

$$a\left[\begin{array}{l}1 \\ 2 \\ 3 \\ 4 \\ 5\end{array}\right]+b\left[\begin{array}{c}-1 \\ 2 \\ -3 \\ 4 \\ -5\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0 \\ 0 ...

GO Classes

638 views

GO Classes asked Mar 14, 2023

9 votes

4 answers

248

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 9
Given a set of vectors $\text{S}(|\text{S}| \geq n)$, with all vectors in $\mathbb{R}^n$, which of the following is a necessary and sufficient condition for the vectors of $\text{S}$ to be Linearly ... $u$ can be represented as a linear combination of vectors (other than $u$ ) of the set $\text{S}$.

Given a set of vectors $\text{S}(|\text{S}| \geq n)$, with all vectors in $\mathbb{R}^n$, which of the following is a necessary and sufficient condition for the vectors o...

GO Classes

747 views

GO Classes asked Mar 14, 2023

20 votes

6 answers

249

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 10
Which of the following is(are) sufficient argument(s) to show that the vectors of set $\text{S}$ ... $7 u+(-1) v+1 w=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$

Which of the following is(are) sufficient argument(s) to show that the vectors of set $\text{S}$ are linearly dependent?$$\text{S}=\left\{u=\left[\begin{array}{c}1 \\-2 \...

GO Classes

1.4k views

GO Classes asked Mar 14, 2023

16 votes

5 answers

250

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 & -...

GO Classes

1.1k views

GO Classes asked Mar 14, 2023

8 votes

3 answers

251

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 12
Given a set $\text{S}$ of vectors in $\mathbb{R}^n$, and set $\text{X(X} \subset \text{S})$ and $\text{Y(S} \subset \text{Y}),$ mark all the statements which are always true: If the ... . If the vectors of $\text{S}$ are Linearly Independent, then the vectors of $\text{Y}$ are also Linearly Independent.

Given a set $\text{S}$ of vectors in $\mathbb{R}^n$, and set $\text{X(X} \subset \text{S})$ and $\text{Y(S} \subset \text{Y}),$ mark all the statements which are always t...

GO Classes

589 views

GO Classes asked Mar 14, 2023

9 votes

2 answers

252

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 14
Which of the following is/are CORRECT? $u,v, \text{ and } w$ are vectors in $\mathbb{R}^n$. A set $\{u, v, w\}$ is linearly independent if $ u $ can not be written as linear combination of $v$ ... $\{\mathbf{u}, \mathbf{v}, \mathbf{w}\}$ is linearly independent.

Which of the following is/are CORRECT?$u,v, \text{ and } w$ are vectors in $\mathbb{R}^n$.A set $\{u, v, w\}$ is linearly independent if $ u $ can not be written as linea...

GO Classes

721 views

GO Classes asked Mar 14, 2023

18 votes

2 answers

253

GO Classes CS/DA 2025 | Weekly Quiz 3 | Fundamental Course and Linear Algebra | Question: 15
$\mathrm{S} 1:$ A set of two vectors in $\mathbb{R}^n$ is Linealy dependent if at least one vector is a multiple of the other. $\text{S2}:$ A set of $n$ vectors in $\mathbb{R}^n$ is Linealy independent if and ... $\text{S} 1$ is incorrect $\text{S} 1$ and $\text{S} 2$ both are incorrect

$\mathrm{S} 1:$ A set of two vectors in $\mathbb{R}^n$ is Linealy dependent if at least one vector is a multiple of the other.$\text{S2}:$ A set of $n$ vectors in $\mathb...

GO Classes

964 views

GO Classes asked Mar 14, 2023

4 votes

2 answers

254

TIFR CSE 2023 | Part B | Question: 5
Consider unit vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^{n}$. Let $\mathbf{w}$ be an arbitrary vector in $\mathbb{R}^{n}$ and $\eta$ be a positive real number such that \[ \mathbf{a}^{\mathbf{T}} \mathbf{b} \geq \eta>0 \geq \ ... $\text{(S3)}$ must be true, but statement $\text{(S2)}$ may be false. Statement $\text{(S1)}$ may be false.

Consider unit vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^{n}$. Let $\mathbf{w}$ be an arbitrary vector in $\mathbb{R}^{n}$ and $\eta$ be a positive real number ...

admin

384 views

admin asked Mar 14, 2023

4 votes

0 answers

255

TIFR CSE 2023 | Part A | Question: 1
Let $A$ be a symmetric $3 \times 3$ matrix with real entries. Let $u$ and $v$ be non-zero vectors with real entries such that $A u=2 u$ and $A v=3 v$. From the set of values $\{0,1,-1\}$, which values can the inner product $u^{T} v$ take? $0$ only $1$ only $-1$ only All of the values $0,1$ and $-1$ None of the values $0,1$ and $-1$

Let $A$ be a symmetric $3 \times 3$ matrix with real entries. Let $u$ and $v$ be non-zero vectors with real entries such that $A u=2 u$ and $A v=3 v$. From the set of val...

admin

539 views

admin asked Mar 14, 2023

4 votes

1 answer

256

TIFR CSE 2023 | Part A | Question: 3
$A$ is an $n \times n$ matrix with real-valued entries. Further, there exists a vector $x \neq 0$ such that $A x=0$. Now consider a given vector $b$ in $\mathbb{R}^{n}$. How many possible vectors $z$ exist, so that $A z=b?$ $0$ $1$ $n-1$ $n$ Either $0$ or infinite

$A$ is an $n \times n$ matrix with real-valued entries. Further, there exists a vector $x \neq 0$ such that $A x=0$. Now consider a given vector $b$ in $\mathbb{R}^{n}$. ...

admin

451 views

admin asked Mar 14, 2023

3 votes

1 answer

257

TIFR CSE 2023 | Part A | Question: 5
Suppose $A$ is a $2 \times 2$ matrix such that the sum of the principal diagonal entries of $A$ is $10$ and the sum of the principal diagonal entries of $A^{2}$ is $20$. (For any $2 \times 2$ matrix $B$, the ... $80$ Nonzero, but cannot be uniquely determined from the above data. Cannot be uniquely determined from the above data, and could also be zero.

Suppose $A$ is a $2 \times 2$ matrix such that the sum of the principal diagonal entries of $A$ is $10$ and the sum of the principal diagonal entries of $A^{2}$ is $20$. ...

admin

337 views

admin asked Mar 14, 2023

4 votes

1 answer

258

TIFR CSE 2023 | Part A | Question: 15
Consider the $n \times n$ matrix $M$ ... $S$ have? $1$ $\left(\begin{array}{l}n \\ 2\end{array}\right)$ $n$ ! $(n !)^{2}$ $n$

Consider the $n \times n$ matrix $M$ defined as follows:$$M=\left(\begin{array}{cccc}1 & 2 & \ldots & n \\n+1 & n+2 & \ldots & 2 n \\2 n+1 & 2 n+2 & \ldots & 3 n \\\vdots...

admin

542 views

admin asked Mar 14, 2023

0 votes

0 answers

259

LU decomposition
Use LU Decomposition method to solve the following system. $\left\{\begin{matrix} & x_{1} +x_{2}-x_{3} =1 \\ & x_{1} +2x_{2}-2x_{3} =0 \\ & -2x_{1} +x_{2}+x_{3} =1 \end{matrix}\right.$ ...

Use LU Decomposition method to solve the following system.$\left\{\begin{matrix} & x_{1} +x_{2}-x_{3} =1 \\ & x_{1} +2x_{2}-2x_{3} =0 \\ & -2x_{1} +x_{2}+x_{3} =1 \end{ma...

kidussss

387 views

kidussss asked Mar 7, 2023

0 votes

0 answers

260

Numerical Method Analysis : Help...
Solve the following system using Gauss elimination with partial pivoting. $\left\{\begin{matrix} &2x_{1}+x_{2}+x_{3}=10\\ & 3x_{1}+2x_{2}+3x_{3}=18 \\ & 5x_{1}+4x_{2}+2x_{3}=9 \end{matrix}\right.$ ...

Solve the following system using Gauss elimination with partial pivoting.$\left\{\begin{matrix} &2x_{1}+x_{2}+x_{3}=10\\ & 3x_{1}+2x_{2}+3x_{3}=18 \\ & 5x_{1}+4x_{2}+2x_{...

kidussss

245 views

kidussss asked Mar 7, 2023

21 votes

4 answers

261

GATE CSE 2023 | Question: 8
Let \[ A=\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{array}\right] \] and \[ B=\left[\begin{array}{llll} 3 & 4 & ... $\operatorname{det}(B)=-\operatorname{det}(A)$ $\operatorname{det}(A)=0$ $\operatorname{det}(A B)=\operatorname{det}(A)+\operatorname{det}(B)$

Let\[A=\left[\begin{array}{llll}1 & 2 & 3 & 4 \\4 & 1 & 2 & 3 \\3 & 4 & 1 & 2 \\2 & 3 & 4 & 1\end{array}\right]\]and\[B=\left[\begin{array}{llll}3 & 4 & 1 & 2 \\4 & 1 & 2...

admin

11.4k views

admin asked Feb 15, 2023

9 votes

4 answers

262

GATE CSE 2023 | Question: 20
Let $A$ be the adjacency matrix of the graph with vertices $\{1,2,3,4,5\}.$ Let $\lambda_{1}, \lambda_{2}, \lambda_{3}, \lambda_{4}$, and $\lambda_{5}$ be the five eigenvalues of $A$. Note that these eigenvalues need not be distinct. The value of $\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}+\lambda_{5}=$____________

Let $A$ be the adjacency matrix of the graph with vertices $\{1,2,3,4,5\}.$Let $\lambda_{1}, \lambda_{2}, \lambda_{3}, \lambda_{4}$, and $\lambda_{5}$ be the five eigenva...

admin

13.2k views

admin asked Feb 15, 2023

4 votes

2 answers

263

GATE CSE 2023 | Memory Based Question: 13
Let $ A=\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{array}\right] $ ... $\operatorname{det} \mathrm{B}=-\operatorname{det} \mathrm{A}$ $\operatorname{det} \mathrm{A}=\operatorname{det} \mathrm{B}$

Let$$A=\left[\begin{array}{llll}1 & 2 & 3 & 4 \\4 & 1 & 2 & 3 \\3 & 4 & 1 & 2 \\2 & 3 & 4 & 1\end{array}\right]$$And$$B=\left[\begin{array}{llll}3 & 4 & 1 & 2 \\4 & 1 & 2...

GO Classes

2.5k views

GO Classes asked Feb 5, 2023

0 votes

0 answers

264

Linear Algebra, Vectors
What is the equation of the plane that contains point (-2, 4, 5) and the vector (7, 0, -6) is normal to the plane? And check if this plane intersects the y-axis.

What is the equation of the plane that contains point (-2, 4, 5) and the vector (7, 0, -6) is normal to the plane? And check if this plane intersects the y-axis.

kidussss

311 views

kidussss asked Jan 13, 2023

0 votes

0 answers

265

Linear Algebra, Vectors
Find equation of a line passes through the points = (0, 1, 2) and = (-1, 1, 1).

Find equation of a line passes through the points = (0, 1, 2) and = (-1, 1, 1).

kidussss

299 views

kidussss asked Jan 13, 2023

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