Recent questions tagged vectors

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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 ... close 0 answers 5 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 Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 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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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 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 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 ...
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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 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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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 $\mathbb{R}^{4}$ ?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 $\mathbb{R}^{4}$ ? For each of th ...
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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)\ ...
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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 ...
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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; ...
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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}$are orthogonal if and only if their direction cosines satisfy 0 answers 9 views Show that two nonzero vectors$\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 Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If$\|\mathbf{v}\|=2$and$\|\mathbf{w}\|=3$, what are the largest and smallest values possible for$\|\mathbf{v}-\mathbf{w}\|$? Give a geometric explanation of your results. 0 answers 12 views If$\|\mathbf{v}\|=2$and$\|\mathbf{w}\|=3$, what are the largest and smallest values possible for$\|\mathbf{v}-\mathbf{w}\|$? Give a geometric explanation of your results.If$\|\mathbf{v}\|=2$and$\|\mathbf{w}\|=3$, what are the largest and smallest values possible for$\|\mathbf{v}-\mathbf{w}\|$? Give a geometric exp ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Are these statements True or False? 0 answers 11 views 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 ... close 0 answers 102 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 ...
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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? ...
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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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Show that $\mathbf{u}=(-2,3,1,4)$ and $\mathbf{v}=(1,2,0,-1)$ are orthogonal vectors in $R^{4}$.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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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.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 ortho ...
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