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Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\).
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Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\).Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\). Prove that \(\operatorname{rank} A+\o ...
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asked in Mathematics 7 minutes ago
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Let \(L: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that
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Let \(L: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show thatLet \(L: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that \ \operatorname{dim} \operatorname{ker}(L)-\operatorname{dim}\left(\o ...
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asked in Mathematics 22 minutes ago
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Find an orthogonal basis for \(\mathcal{S}\) and use it to find the \(3 \times 3\) matrix \(P\) that projects vectors orthogonally into \(\mathcal{S}\).

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Let \(\mathcal{P}_{3}\) be the space of polynomials of degree at most 3 anD let \(D: \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}\) be the derivative operator.

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Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 .
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Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 .Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 .   a) Find a basis for this space. b) Let \(D: \mathcal{P}_{2} \righta ...
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asked in Mathematics 35 minutes ago
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Answer the following in terms of \(\mathbf{V}, \mathbf{W}\), and \(\mathbf{Z}\).
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Answer the following in terms of \(\mathbf{V}, \mathbf{W}\), and \(\mathbf{Z}\).Let \(A\) be a matrix, not necessarily square. Say \(\mathbf{V}\) and \(\mathbf{W}\) are particular solutions of the equations \(A \mathbf{V}=\mathbf{ ...
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asked in Mathematics 44 minutes ago
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If \(A\) and \(B\) are \(4 \times 4\) matrices such that \(\operatorname{rank}(A B)=3\), then \(\operatorname{rank}(B A)<4\).
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If \(A\) and \(B\) are \(4 \times 4\) matrices such that \(\operatorname{rank}(A B)=3\), then \(\operatorname{rank}(B A)<4\).For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE. There is no need to ju ...
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asked in Mathematics 56 minutes ago
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a real matrix, not necessarily square.
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a real matrix, not necessarily square.Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a real matrix, not necessarily square. a) If two columns of \(A\) are the same, show that \(A ...
by MathsGee Platinum
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asked in Mathematics 9 hours ago
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that the following are equivalent.
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that the following are equivalent.Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that the following are equivalent. a) For every \(y \in \mathbb{R}^{k}\) th ...
by MathsGee Platinum
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asked in Mathematics 12 hours ago
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For which real numbers \(x\) do the vectors: \((x, 1,1,1),(1, x, 1,1),(1,1, x, 1),(1,1,1, x)\) not form a basis of \(\mathbb{R}^{4}\) ?
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Which of the following sets of vectors are bases for \(\mathbb{R}^{2} ?\)
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Which of the following sets of vectors are bases for \(\mathbb{R}^{2} ?\)Which of the following sets of vectors are bases for \(\mathbb{R}^{2} ?\) a). \(\{(0,1),(1,1)\}\) d). \(\{(1,1),(1,-1)\}\) b). \(\{(1,0),(0,1),(1,1)\ ...
by MathsGee Platinum
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asked in Mathematics 13 hours ago
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Use Head to Tail Method to represent the following vectors and finally draw Resultant vector.
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Use Head to Tail Method to represent the following vectors and finally draw Resultant vector.F1=12N in the positive x direction. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;F2=10N in the positive x direction. &nbsp;&nbs ...
by Khensanikesty
(104 points)
asked in Physics 2 days ago
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How do ISBN numbers apply dot products?
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How do ISBN numbers apply dot products? How do ISBN numbers apply dot products? &nbsp; ...
by MathsGee Platinum
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asked in Mathematics Jan 13
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Show that two nonzero vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in $R^{3}$ are orthogonal if and only if their direction cosines satisfy
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Are these statements True or False?
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Are these statements True or False?(a) If each component of a vector in $R^{3}$ is doubled, the norm of that vector is doubled. &nbsp; (b) In $R^{2}$, the vectors of norm 5 whose init ...
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asked in Mathematics Jan 13
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When are two nonzero vectors orthogonal to each other?
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When are two nonzero vectors orthogonal to each other?When are two nonzero vectors orthogonal to each other? ...
by MathsGee Platinum
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asked in Mathematics Jan 13
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Show that $\mathbf{u}=(-2,3,1,4)$ and $\mathbf{v}=(1,2,0,-1)$ are orthogonal vectors in $R^{4}$.
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Let $S=\{\mathbf{i}, \mathbf{j}, \mathbf{k}\}$ be the set of standard unit vectors in $R^{3} .$ Show that each ordered pair of vectors in $S$ is orthogonal.

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