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Syllabus: Matrices, determinants, System of linear equations, Eigenvalues and eigenvectors, LU decomposition.

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2024-1} & \textbf{2024-2} & \textbf{2023} & \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} &1&0&2& 1 &0&1&0&0.83&2
\\\hline\textbf{2 Marks Count} &1&1&0& 2 &1&1&0&1&2
\\\hline\textbf{Total Marks} &3&3&2& 5 &2&3&\bf{2}&\bf{3}&\bf{5}\\\hline
\end{array}}}$$

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Go classes linear algebra lecture filling a vector space.
Why does linear combination of 2 linearly independent vectors produce every vector in R^2 ?
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177
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Go Classes LA for GATE DA | Homework: 2 | question: 6
Q: The given matrix has solution for:$\begin{bmatrix} 1 & 1 & 3\\ 1 & 2 & 5\\ 2 & 3 & 8 \end{bmatrix}$a. All vectors b in $\mathbb{R}^{3}$ ... ?2) why option D is incorrect ?
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119
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Self doubt
1. If v1 and v2 are linearly independent eigenvectors then they can correspond to the same eigenvalue.2. λ is the eigenvalue of A if and only if λ is the eigenvalue of A transpose.Can anyone please explain these two statements
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162
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ISI PCA 2024 | Question : 2
Let $\mathbf{C} = \{ (1, 2), (2, 1) \}$ be a basis of $\mathbb{R}^2$ and $T: \mathbb{R}^2 \to \mathbb{R}^2$ be defined by\[T \begin{pmatrix} x \\ y \end{ ... $T(C) = \begin{pmatrix} 3 & -1 \\ -3 & 2 \end{pmatrix}$ 
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self doubt
For the system of linear equation Ax=0 where matrix A(mxn) , what can we say about the number of solutions for this equation :1. if all n columns of A are linearly Independent.2. if less than n columns of A are linearly Independent.
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164
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IIT Madras MS DSAI Written Test 2024
Given, that the eigen values of a 2 x 2 matrix are -1,1 and its singular values are 1,0. What is the rank of the matrix?a) rank is 0b) rank is 1c) Such a matrix can't existd) rank is 2
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152
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4 votes
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 $\left(T_{2}\right)=3$ ... s.t $T_{3}\left(T_{1}\right)=T_{2}$. Then find rank of $T_{3}$
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132
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3 votes
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:0123
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129
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1 answers
3 votes
GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 4
Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\}$ ... 2 \\ 1\end{array}\right]$\left[\begin{array}{r}3 \\ -1 \\ 1\end{array}\right]$
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147
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2 answers
5 votes
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 $\mathcal{E}$ ... -1\end{array}\right)$\left(\begin{array}{ll}0 & -1 \\ 1 & -1\end{array}\right)$
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117
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GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 6
Suppose that ...
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158
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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 $A^{4} \mathbf{v}=\mathbf{0}$ ... $\mathcal{B}$ is a basis of $\mathbb{R}^{4}$.$\mathcal{B}$ is not linearly independent.
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132
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1 answers
1 votes
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 $\operatorname{Rank}(A)=2$.How many solutions ... infinitely many solutions, (ii) no solution.(i) infinitely many solutions, (ii) unique solution.
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117
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1 answers
2 votes
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 $c$ ... 0$4 a-3 b+c \neq 0$3 a-4 b-c \neq 0$4 a-3 b-c \neq 0$
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118
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1 answers
1 votes
GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 10
Let $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ ... )$\left(\begin{array}{lll}2 & 1 & 3 \\ 6 & 3 & 9\end{array}\right)$
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108
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1 answers
3 votes
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 ... rank $n$, then the column space of $A$ is equal to the column space of $B$.
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163
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2 answers
3 votes
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 $V$ ... , $\mathrm{S} 2$ is true.Both S1 and S2 are true.Both S1 and S2 are false.
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131
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2 answers
5 votes
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 ... has domain $\mathbf{R}^{2}$ and codomain $\mathbf{R}^{4}$.$T$ cannot be onto.
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143
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1 answers
2 votes
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 $T(x)=A x$. Which of the following statements are ... $\operatorname{rank}(A) \leq 4$.
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109
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GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 15
The matrix$\left(\begin{array}{rr}-2 & 11 \\4 & 2\end{array}\right)$represents a linear transformation $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ with respect to ... }\right)$\left(\begin{array}{rr}-2 & 11 \\ 4 & 2\end{array}\right)$
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124
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1 answers
4 votes
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 $T(x)=A x$. Which of the following statements are ... $\operatorname{rank}(A) \leq 4$.
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361
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2 answers
6 votes
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 ... is $x = 0$. The rank of $A$ is $n$.
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415
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4 answers
8 votes
GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 2
Let $M$ be a $2 \times 2$ matrix with the property that the sum of each of the rows and also the sum of each of the columns is the same constant $c$. Which (if any) ... [\begin{array}{l}1 \\ 1 \\ \end{array}\right]$U$V$W$None of the above
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463
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1 answers
4 votes
GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 3
Let the $n \times n$ matrix $A$ have an eigenvalue $\lambda$ with corresponding eigenvector $v$.Which of the following statements are true for matrix $A$.$-v$ is an ... $A^3$ is $\lambda^3$ and the eigenvector is $v^3$ .
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307
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3 answers
2 votes
GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 4
Consider the following matrix A:$\left[\begin{array}{lll}2 & -1 & 0 \\ 0 & 2 & 0 \\ 1 & 0 & 2\end{array}\right]$ ... to eigenvalue $\lambda$ of $A$ then $x$ is also the eigenvector of $A^{-1}$.
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175
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1 answers
1 votes
GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 5
Suppose a $3 \times 5$ matrix $A$ has rank $r = 3$. Then the equation $Ax = b$ $\textbf{BLANK 1}$  has  $\textbf{BLANK 2}$. ... many solutionsBLANK 1: Sometimes, BLANK 2: Unique solutionBLANK 1: Sometimes, BLANK 2: Infinitely many solutions
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273
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3 answers
4 votes
GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 6
Which of the following statements is/are $\textbf{NOT CORRECT}$?If $v1$ and $v2$ are linearly independent eigenvectors then they can correspond to the same eigenvalue.If $A$ is a ... $A$ then $\lambda ^{-1}$ is an eigenvalue of $A^{-1}$.
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202
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1 answers
4 votes
GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 7
Given an $m \times n$ matrix $A$ whose rows are linearly independent. Now, consider following statements regarding $A$:  $S1:$ The system of equations $Ax = b$ ... FALSE.$S1$ is FALSE and $S2$ is TRUE.Both $S1$ and $S2$ are FALSE.
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310
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1 answers
4 votes
GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 8
Which of the following statements is/are $\textbf{FALSE}$?For $n \times n$ real-symmetric matrices $A$ and $B$, $AB$ and $BA$ always have the same eigenvalues ... matrices $A$ and $B$, $AB$ and $BA$ always have the same eigenvectors.
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207
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1 answers
5 votes
GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 9
Let $A$ and $B$ be two $n \times n$ matrices. If $B$ is invertible and $(I+BA)^{-1} = 2B^2$, then which of the following is the correct definition of $A$ in terms of $B$ ... $A = 2B^3-I$None of the above
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