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Recent questions tagged orthogonal


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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 \(\mathcal{S} \subset \mathbb{R}^{4}\) be the vectors \(X=\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\) that satisfy \(x_{1}+x_{2}-x_{3}+x_{4}=0\).
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Find the (orthogonal) projection of \(\mathbf{x}:=(1,2,0)\) into the following subspaces:
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Find the (orthogonal) projection of \(\mathbf{x}:=(1,2,0)\) into the following subspaces:Find the (orthogonal) projection of \(\mathbf{x}:=(1,2,0)\) into the following subspaces: a) The line spanned by \(\mathbf{u}:=(1,1,-1)\). b) The plan ...
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Proof or counterexample. Here \(v, w, z\) are vectors in a real inner product space \(H\).
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Proof or counterexample. Here \(v, w, z\) are vectors in a real inner product space \(H\).Proof or counterexample. Here \(v, w, z\) are vectors in a real inner product space \(H\). a) Let \(v, w, z\) be vectors in a real inner product space ...
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Let \(U, V, W\) be orthogonal vectors and let \(Z=a U+b V+c W\), where \(a, b, c\) are scalars.
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Let \(U, V, W\) be orthogonal vectors and let \(Z=a U+b V+c W\), where \(a, b, c\) are scalars.Let \(U, V, W\) be orthogonal vectors and let \(Z=a U+b V+c W\), where \(a, b, c\) are scalars. a) (Pythagoras) Show that \(\|Z\|^{2}=a^{2}\|U\|^{2}+b ...
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In \(\mathbb{R}^{3}\), let \(N\) be a non-zero vector and \(X_{0}\) and \(Z\) points.
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In \(\mathbb{R}^{3}\), let \(N\) be a non-zero vector and \(X_{0}\) and \(Z\) points.In \(\mathbb{R}^{3}\), let \(N\) be a non-zero vector and \(X_{0}\) and \(Z\) points. a) Find the equation of the plane through the origin that is ort ...
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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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Find an orthogonal matrix \(R\) that diagonalizes \(A:=\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)\)
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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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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 \(\mathcal{P}_{2}\) 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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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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Find a matrix that rotates the plane through \(+60\) degrees, keeping the origin fixed.
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Find a \(2 \times 2\) matrix that rotates the plane by \(+45\) degrees followed by a reflection across the horizontal axis.
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