# Recent questions tagged orthogonal

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Let $V \subset \mathbb{R}^{n}$ be a linear space, $Q: R^{n} \rightarrow V^{\perp}$ the orthogonal projection into $V^{\perp}$, and $x \in \mathbb{R}^{n}$ a given vector.Dual variational problems Let $V \subset \mathbb{R}^{n}$ be a linear space, $Q: R^{n} \rightarrow V^{\perp}$ the orthogonal projection into $V^ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 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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Using the inner product of the previous problem, let $\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}$ be an orthogonal basis for the space $\mathcal{P}_{2}$ of quadratic polynomials and . . .Using the inner product of the previous problem, let $\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}$ be an orthogonal basis for the space $\mathcal{P} ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Using the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x$, for which values of the real constants $\alpha, \beta, \gamma$ are the quadratic polynomials ...
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Using the inner product $\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x$, for which values of the real constants $\alpha, \beta, \gamma$ are the quadratic polynomials ...Using the inner product $\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x$, for which values of the real constants $\alpha, \beta, \gamma$ are the ...
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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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Let $\mathcal{S} \subset \mathbb{R}^{4}$ be the vectors $X=\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ that satisfy $x_{1}+x_{2}-x_{3}+x_{4}=0$.Let $\mathcal{S} \subset \mathbb{R}^{4}$ be the vectors $X=\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ that satisfy $x_{1}+x_{2}-x_{3}+x_{4}=0$. a) ...
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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 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 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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Find all vectors in the plane (through the origin) spanned by $\mathbf{V}=(1,1-2)$ and $\mathbf{W}=(-1,1,1)$ that are perpendicular to the vector $\mathbf{Z}=(2,1,2)$.
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Find all vectors in the plane (through the origin) spanned by $\mathbf{V}=(1,1-2)$ and $\mathbf{W}=(-1,1,1)$ that are perpendicular to the vector $\mathbf{Z}=(2,1,2)$.Find all vectors in the plane (through the origin) spanned by $\mathbf{V}=(1,1-2)$ and $\mathbf{W}=(-1,1,1)$ that are perpendicular to the vector ...
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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 $A$ be an $m \times n$ matrix, and suppose $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^{T} A$. Show that $A \vec{v}$ and $A \vec{w}$ are orthogonal.Let $A$ be an $m \times n$ matrix, and suppose $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^{T} A$. Show that $A \vec{v}$ and ...
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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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Find an orthogonal matrix $R$ that diagonalizes $A:=\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)$Find an orthogonal matrix $R$ that diagonalizes $A:=\left(\begin{array}{rrr}1 &amp; -1 &amp; 0 \\ -1 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 2\end{array ... close 0 answers 7 views Let \(\vec{v}$ and $\vec{w}$ be vectors in $\mathbb{R}^{n}$. If $\|\vec{v}\|=\|\vec{w}\|$, show there is an orthogonal matrix $R$ with $R \vec{v}=\vec{w}$ and $R \vec{w}=\vec{v}$.Let $\vec{v}$ and $\vec{w}$ be vectors in $\mathbb{R}^{n}$. If $\|\vec{v}\|=\|\vec{w}\|$, show there is an orthogonal matrix $R$ with $R \v ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 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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If $\mathbf{n}=(a, b, c)$ is a unit vector, use this formula to show that (perhaps surprisingly) the orthogonal projection of $\mathbf{x}$ into the plane perpendicular to $\mathbf{n}$ is given bya) Let $\mathbf{v}:=(a, b, c)$ and $\mathbf{x}:=(x, y, z)$ be any vectors in $\mathbb{R}^{3}$. Viewed as column vectors, find a $3 \times 3$ m ...
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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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Find an orthogonal basis for $\mathcal{S}$ and use it to find the $3 \times 3$ matrix $P$ that projects vectors orthogonally into $\mathcal{S}$.Let $\mathcal{S} \subset \mathbb{R}^{3}$ be the subspace spanned by the two vectors $v_{1}=(1,-1,0)$ and $v_{2}=$ $(1,-1,1)$ and let $\mathca ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 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 $\mathcal{P}_{3}$ be the space of polynomials of degree at most 3 anD let $D: \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}$ be the derivative operator.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 o ...
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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 . &nbsp; a) Find a basis for this space. b) Let $D: \mathcal{P}_{2} \righta ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(\mathcal{P}_{1}$ be the linear space of real polynomials of degree at most one, so a typical element is $p(x):=a+b x$, where $a$ and $b$ are real numbers.
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Let $\mathcal{P}_{1}$ be the linear space of real polynomials of degree at most one, so a typical element is $p(x):=a+b x$, where $a$ and $b$ are real numbers.Let $\mathcal{P}_{1}$ be the linear space of real polynomials of degree at most one, so a typical element is $p(x):=a+b x$, where $a$ and $b$ ...
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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{ ... close 0 answers 4 views Linear maps \(F(X)=A X$, where $A$ is a matrix, have the property that $F(0)=A 0=0$, so they necessarily leave the origin fixed. It is simple to extend this to include a translation, Linear maps $F(X)=A X$, where $A$ is a matrix, have the property that $F(0)=A 0=0$, so they necessarily leave the origin fixed. It is simple t ...
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Find a $3 \times 3$ matrix $A$ mapping $\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ that rotates the $x_{1} x_{3}$ plane by 60 degrees and leaves the $x_{2}$ axis fixed.Find a $3 \times 3$ matrix $A$ mapping $\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ that rotates the $x_{1} x_{3}$ plane by 60 degrees and leav ...
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Find a matrix that rotates the plane through $+60$ degrees, keeping the origin fixed.
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Find a matrix that rotates the plane through $+60$ degrees, keeping the origin fixed.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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Find a $2 \times 2$ matrix that rotates the plane by $+45$ degrees followed by a reflection across the horizontal axis.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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