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Recent questions tagged vector-space
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Memory Based GATE DA 2024 | Question: 5
Consider a matrix \(M \in \mathbb{R}^{3 \times 3}\) and let \(U\) be a 2-dimensional subspace such that \(M\) is a projection onto \(U\). Which of the following statements are true? \(M^3 = M\) \(M^2 = M\) The nullspace of \(M\) is 1-dimensional. The nullspace of \(M\) is 2-dimensional.
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Memory Based GATE DA 2024 | Question: 29
Consider the vector \( u = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{bmatrix} \), and let \( M = uu^{\top} \). If \( \sigma_1, \sigma_2, \sigma_3, \ldots, \sigma_5 \) are the singular values of \( M \), what is the value of \( \sum_{i=1}^5 \sigma_i \)?
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Memory Based GATE DA 2024 | Question: 60
Linear Algebra Question: Four options were given related to subspace R3. Something like this : A. \( \alpha \cdot x + \beta \cdot y \) B. \( \alpha^2 \cdot x + \beta^2 \cdot y \) C. \(f(x) = 4x_1 + 2x_3 + 3x_3 \) D.
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GATE 2016 | MATHS | Q-11
Let \( \mathbf{v}, \mathbf{w}, \mathbf{u} \) be a basis of \( \mathbb{V} \). Consider the following statements P and Q: (P) : \( \{\mathbf{v} + \mathbf{w}, \mathbf{w} + \mathbf{u}, \mathbf{v} - \mathbf{u}\} \) is a basis of \( \mathbb{V} \). ( ... a basis of \( \mathbb{V} \). Which of the above statements hold TRUE? (A) both P and Q (B) only P (C) only Q (D) Neither P nor Q
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Jan 11
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GATE 2018 | MATHS | Q-51
Consider \( \mathbb{R}^3 \) with the usual inner product. If \( d \) is the distance from \( (1, 1, 1) \) to the subspace ${(1, 1, 0), (0, 1, 1)}$ of \( \mathbb{R}^3 \), then \( 3d^2 \) is given by
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GATE 2018 | MATHS | Q-50
Let \( M_2(\mathbb{R}) \) be the vector space of all \( 2 \times 2 \) real matrices over the field \( \mathbb{R} \). Define the linear transformation \( S : M_2(\mathbb{R}) \to M_2(\mathbb{R}) \) by \( S(X) = 2X + X^T \), where \( X^T \) denotes the transpose of the matrix \( X \). Then the trace of \( S \) equals________
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GATE 2019 | Maths | DA Sample questions
Let $V$ be the vector space of all $3 \times 3$ matrices with complex entries over the real field. If $W_1 = \{A \in V : A = \bar{\mathbf{A}}^T \}$ and $W_2 = \{A \in V : trace(A)=0\}$, then the dimension of $W_1 + W_2$ is equal to ______________ ($\bar{\mathbf{A}}^T $ denotes the conjugate transpose of $A$.)
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Jan 10
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GATE 2021 | MATHS | PRACTICE PROBLEMS FOR DA PAPER
Let $ \langle \cdot, \cdot \rangle: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} $ be an inner product on the vector space $ \mathbb{R}^n $ over $ \mathbb{R} $. Consider the following statements: $P:$ ... P and Q are TRUE (B) P is TRUE and Q is FALSE (C) P is FALSE and Q is TRUE (D) both P and Q are FALSE
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GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 18
In this problem, consider the $4 \times 4$ matrix $A$ whose columns are vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, \mathbf{a}_4 \in \mathbb{R}^4$ ... a element in matrix $\text{A}$ at $\mathrm{i}^{\text {th }}$ row and $\mathrm{j}^{\text{th}}$ column.
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GO Classes Weekly Quiz 6 | Linear Algebra | Question: 1
Consider the vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ ... independent otherwise The vectors $\left\{\mathbf{v}_1, \mathbf{v}_2\right\}$ are linearly independent when $t=4$, and linearly dependent otherwise
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GO Classes Weekly Quiz 6 | Linear Algebra | Question: 9
$ \left[\begin{array}{ll} 3 & 1 \\ 1 & 0 \\ 2 & 5 \end{array}\right]\left[\begin{array}{lll} a & 1 & 0 \\ 2 & b & 1 \end{array}\right]=A_{3 \times 3} $ At what values of $(a, b), A_{3 \times 3}$ will be invertible? $(1,-1)$ $(-1,1)$ Both A and B None of the above
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GO Classes Weekly Quiz 6 | Linear Algebra | Question: 17
Let $A$ be an $m \times n$-matrix and let $B$ be an $n \times m$-matrix. Then which of the following statement is not true for all such matrices? $B A$ is defined the columns of $A B$ are linear combinations of the columns of $B$ $A B$ is defined the columns of $A B$ are linear combinations of the columns of $A$
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GO Classes Weekly Quiz 6 | Linear Algebra | Question: 18
Consider $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 0\end{array}\right]$ ... $B$. $\boldsymbol{v}$ is neither a linear combination of the columns of $A$ nor of the columns of $B$.
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GO Classes Weekly Quiz 6 | Linear Algebra | Question: 27
Consider the following set of (column) vectors: $X=\left\{\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right] \in \mathbb{R}^3 \mid 2 x_1+3 x_2-x_3=0\right\}.$ Which of the following statements are true? Every element of ... Choose the correct option. Only I is true. Only II is true. Both I and II are true. Neither I, nor II are true.
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