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Let \(A\) be a positive definite \(n \times n\) real matrix, \(\vec{b}\) a real vector, and \(\vec{N}\) a real unit vector.
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Let \(A\) be a positive definite \(n \times n\) real matrix, \(\vec{b}\) a real vector, and \(\vec{N}\) a real unit vector.Let \(A\) be a positive definite \(n \times n\) real matrix, \(\vec{b}\) a real vector, and \(\vec{N}\) a real unit vector. a) For which value(s) of t ...
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Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.
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Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials. a) Show that \(\langle f, g\rangle=f(-1) g(-1)+f(0) g(0)+f(1) g(1)\) is an inner produ ...
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map defined by the matrix \(A\).
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map defined by the matrix \(A\).Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map defined by the matrix \(A\). If the matrix \(B\) satisfies the relation \(\langle ...
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Let \(\ell\) be any linear functional. Show there is a unique vector \(v \in \mathbb{R}^{n}\) so that \(\ell(x):=\langle x, v\rangle\).
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Let \(V, W\) be vectors in \(\mathbb{R}^{n}\).
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Let \(V, W\) be vectors in \(\mathbb{R}^{n}\).Let \(V, W\) be vectors in \(\mathbb{R}^{n}\). a) Show that the Pythagorean relation \(\|V+W\|^{2}=\|V\|^{2}+\|W\|^{2}\) holds if and only if \(V\) a ...
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Let \(U \subset V\) and \(W\) be finite dimensional linear spaces and \(L: V \rightarrow W\) a linear map. Show that
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Let \(U \subset V\) and \(W\) be finite dimensional linear spaces and \(L: V \rightarrow W\) a linear map. Show thatLet \(U \subset V\) and \(W\) be finite dimensional linear spaces and \(L: V \rightarrow W\) a linear map. Show that \ \operatorname{dim}\left(\left. ...
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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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Let \(L: V \rightarrow V\) be a linear map on a vector space \(V\).
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Let \(L: V \rightarrow V\) be a linear map on a vector space \(V\).Let \(L: V \rightarrow V\) be a linear map on a vector space \(V\). a) Show that \(\operatorname{ker} L \subset \operatorname{ker} L^{2}\) and, more g ...
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Which of the following are not a basis for the vector space of all symmetric \(2 \times 2\) matrices? Why?
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Which of the following are not a basis for the vector space of all symmetric \(2 \times 2\) matrices? Why?Which of the following are not a basis for the vector space of all symmetric \(2 \times 2\) matrices? Why?   a) \(\left(\begin{array}{ll}1 &amp ...
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Give an example of a linear transformation \(L: V \rightarrow V\) (or show that there is no such transformation) for which:
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Give an example of a linear transformation \(L: V \rightarrow V\) (or show that there is no such transformation) for which:Let \(V\) be a vector space with \(\operatorname{dim} V=10\) and let \(L: V \rightarrow V\) be a linear transformation. Consider \(L^{k}: V \rightarro ...
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Find the dimension of \(A\) considered as a real vector space.
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Find the dimension of \(A\) considered as a real vector space.Consider the two linear transformations on the vector space \(V=\mathbf{R}^{n}\) : \(R=\) right shift: \(\left(x_{1}, \ldots, x_{n}\right) \rightarrow ...
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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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Show that \(\mathcal{L}\) and \(\mathcal{R}\) are linear spaces and compute their dimensions.
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Show that \(\mathcal{L}\) and \(\mathcal{R}\) are linear spaces and compute their dimensions.Say \(A \in M(n, \mathbb{F})\) has rank \(k\). Define \ \mathcal{L}:=\{B \in M(n, \mathbb{F}) \mid B A=0\} \quad \text { and } \quad \mathcal{R}:=\{C ...
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