# Recent questions tagged product

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Let $V$ be the real vector space of continuous real-valued functions on the closed interval $[0,1]$, and let $w \in V$. For $p, q \in V$, define $\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x$.Let $V$ be the real vector space of continuous real-valued functions on the closed interval $0,1$, and let $w \in V$. For $p, q \in V$, defi ...
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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.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 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 $\mathbf{Z}=(2,1,2)$.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 $\mathbb{R}^{n}$.
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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Let $W$ be a linear space with an inner product and $A: W \rightarrow W$ be a linear map whose image is one dimensional (so in the case of matrices, it has rank one).
Let $W$ be a linear space with an inner product and $A: W \rightarrow W$ be a linear map whose image is one dimensional (so in the case of matrices, it has rank one).Let $W$ be a linear space with an inner product and $A: W \rightarrow W$ be a linear map whose image is one dimensional (so in the case of matrice ...
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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 \(\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 .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.
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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If $A$ and $B$ are $4 \times 4$ matrices such that $\operatorname{rank}(A B)=3$, then $\operatorname{rank}(B A)<4$.
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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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.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 ...
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Let $\mathcal{P}_{k}$ be the space of polynomials of degree at most $k$ and define the linear map $L: \mathcal{P}_{k} \rightarrow \mathcal{P}_{k+1}$ by $L p:=p^{\prime \prime}(x)+x p(x) .$Let $\mathcal{P}_{k}$ be the space of polynomials of degree at most $k$ and define the linear map $L: \mathcal{P}_{k} \rightarrow \mathcal{P}_{k+ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What are the rules of differentiation? 1 answer 7 views What are the rules of differentiation?What are the rules of differentiation? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Verify that the Cauchy-Schwarz inequality holds. \(\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$
Verify that the Cauchy-Schwarz inequality holds. $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$Verify that the Cauchy-Schwarz inequality holds. (b) $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$ ...
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What is the dot product of $\mathbf{u}$ a column matrix and $\mathbf{v}$ a column matrix?
What is the dot product of $\mathbf{u}$ a column matrix and $\mathbf{v}$ a column matrix?What is the dot product of $\mathbf{u}$ a column matrix and $\mathbf{v}$ a column matrix? ...
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What is the dot product of u a row matrix and v a column matrix?
What is the dot product of u a row matrix and v a column matrix?What is the dot product of u a row matrix and v a column matrix? ...
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What is the dot product of u a column matrix and v a row matrix?
What is the dot product of u a column matrix and v a row matrix?What is the dot product of u a column matrix and v a row matrix? ...
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Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$
Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$Verify that the Cauchy&amp;ndash;Schwarz inequality holds for $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$ ...
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Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3)$
Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3)$Verify that the Cauchy&amp;ndash;Schwarz inequality holds for $\mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3)$ ...
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Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2)$
Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2)$Verify that the Cauchy&amp;ndash;Schwarz inequality holds for $\mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2)$ ...
Let $\mathbf{r}_{0}=\left(x_{0}, y_{0}\right)$ be a fixed vector in $R^{2}$. In each part, describe in words the set of all vectors $\mathbf{r}=(x, y)$ that satisfy the stated condition.Let $\mathbf{r}_{0}=\left(x_{0}, y_{0}\right)$ be a fixed vector in $R^{2}$. In each part, describe in words the set of all vectors $\mathbf{r}=(x, y) ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that two nonzero vectors$\mathbf{v}_{1}$and$\mathbf{v}_{2}$in$R^{3}$are orthogonal if and only if their direction cosines satisfy 0 answers 13 views Show that two nonzero vectors$\mathbf{v}_{1}$and$\mathbf{v}_{2}$in$R^{3}$are orthogonal if and only if their direction cosines satisfyShow that two nonzero vectors$\mathbf{v}_{1}$and$\mathbf{v}_{2}$in$R^{3}$are orthogonal if and only if their direction cosines satisfy$$\cos \ ... close 0 answers 104 views Let$\mathbf{u}$be a vector in$R^{100}$whose$i$th component is$i$, and let$\mathbf{v}$be the vector in$R^{100}$whose$i$th component is$1 /(i+1)$. Find the dot product of$\mathbf{u}$and$\mathbf{v}$.Let$\mathbf{u}$be a vector in$R^{100}$whose$i$th component is$i$, and let$\mathbf{v}$be the vector in$R^{100}$whose$i$th component is$1 ...