# Recent questions tagged prove

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Let $A$ be a square matrix and let $\|B\|$ be any norm on matrices [one example is $\left.\|B\|=\max _{i, j}\left|b_{i j}\right|\right]$.
Let $A$ be a square matrix and let $\|B\|$ be any norm on matrices [one example is $\left.\|B\|=\max _{i, j}\left|b_{i j}\right|\right]$.Let $A$ be a square matrix and let $\|B\|$ be any norm on matrices one example is $\left.\|B\|=\max _{i, j}\left|b_{i j}\right|\right$. To wha ...
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Let $\alpha_{1}, \ldots, \alpha_{n}$ be positive real numbers and let $A=\left(a_{i j}\right)$ where $a_{i j}=\alpha_{i} / \alpha_{j} .$ Say as much as you can about the eigenvalues and eigenvectors of $A$.Let $\alpha_{1}, \ldots, \alpha_{n}$ be positive real numbers and let $A=\left(a_{i j}\right)$ where $a_{i j}=\alpha_{i} / \alpha_{j} .$ Say as ...
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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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Show that $\operatorname{trace}(A B C)=\operatorname{trace}(C A B)=\operatorname{trace}(B C A)$.
Show that $\operatorname{trace}(A B C)=\operatorname{trace}(C A B)=\operatorname{trace}(B C A)$.Let $A, B$, and $C$ be any $n \times n$ matrices. a) Show that $\operatorname{trace}(A B)=\operatorname{trace}(B A)$. b) Show that $\operator ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Every real upper triangular \(n \times n$ matrix $\left(a_{i j}\right)$ with $a_{i i}=1, i=1,2, \ldots, n$ is invertible. Proof or counterexample.
Every real upper triangular $n \times n$ matrix $\left(a_{i j}\right)$ with $a_{i i}=1, i=1,2, \ldots, n$ is invertible. Proof or counterexample.Every real upper triangular $n \times n$ matrix $\left(a_{i j}\right)$ with $a_{i i}=1, i=1,2, \ldots, n$ is invertible. Proof or counterexample ...
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Let $A$ be the rank one $n \times n$ matrix $A=\left(v_{i} v_{j}\right)$, where $\vec{v}:=\left(v_{1}, \ldots, v_{n}\right)$ is a non-zero real vector.
Let $A$ be the rank one $n \times n$ matrix $A=\left(v_{i} v_{j}\right)$, where $\vec{v}:=\left(v_{1}, \ldots, v_{n}\right)$ is a non-zero real vector.Let $A$ be the rank one $n \times n$ matrix $A=\left(v_{i} v_{j}\right)$, where $\vec{v}:=\left(v_{1}, \ldots, v_{n}\right)$ is a non-zero rea ...
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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 0 answers 3 views 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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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 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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Answer the following in terms of $\mathbf{V}, \mathbf{W}$, and $\mathbf{Z}$.
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 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 1 answer 18 views Find the common ratio of the geometric sequence \(16,24,36,54, \ldots$ Then express each sequence in the form $a_{n}=a_{1} r^{n-1}$ and find the eighth term of the sequence.Find the common ratio of the geometric sequence $16,24,36,54, \ldots$ Then express each sequence in the form $a_{n}=a_{1} r^{n-1}$ and find the ei ...
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Find the common ratio of the geometric sequence $1,3,9,27, \ldots$ Then express each sequence in the form $a_{n}=a_{1} r^{n-1}$ and find the eighth term of the sequence.
Find the common ratio of the geometric sequence $1,3,9,27, \ldots$ Then express each sequence in the form $a_{n}=a_{1} r^{n-1}$ and find the eighth term of the sequence.Find the common ratio of the geometric sequence $1,3,9,27, \ldots$ Then express each sequence in the form $a_{n}=a_{1} r^{n-1}$ and find the eight ...
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What is aleph?
What is aleph?What is aleph? ...
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What is affirming the consequent?
What is affirming the consequent?What is affirming the consequent? ...
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If A is an $n\times n$ matrix and u and v are $n\times 1$ matrices, then prove that $\boldsymbol{A u} \cdot \mathbf{v}=\mathbf{u} \cdot \boldsymbol{A}^{T} \mathbf{v}$
If A is an $n\times n$ matrix and u and v are $n\times 1$ matrices, then prove that $\boldsymbol{A u} \cdot \mathbf{v}=\mathbf{u} \cdot \boldsymbol{A}^{T} \mathbf{v}$If A is an $n\times n$ matrix and u and v are $n\times 1$ matrices, then prove that $\boldsymbol{A u} \cdot \mathbf{v}=\mathbf{u} \cdot \boldsymbol{A} ... close 0 answers 45 views Let$\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) ...
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
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 $$\cos \ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If \|\mathbf{v}\|=2 and \|\mathbf{w}\|=3, what are the largest and smallest values possible for \|\mathbf{v}-\mathbf{w}\| ? Give a geometric explanation of your results. 0 answers 12 views If \|\mathbf{v}\|=2 and \|\mathbf{w}\|=3, what are the largest and smallest values possible for \|\mathbf{v}-\mathbf{w}\| ? Give a geometric explanation of your results.If \|\mathbf{v}\|=2 and \|\mathbf{w}\|=3, what are the largest and smallest values possible for \|\mathbf{v}-\mathbf{w}\| ? Give a geometric exp ... close 0 answers 103 views Let \mathbf{u} be a vector in R^{100} whose i th component is i, and let \mathbf{v} be the vector in R^{100} whose i th component is 1 /(i+1). Find the dot product of \mathbf{u} and \mathbf{v}.Let \mathbf{u} be a vector in R^{100} whose i th component is i, and let \mathbf{v} be the vector in R^{100} whose i th component is 1 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If \mathbf{v} is a vector in R^{n}, and if k is any scalar, then prove that \|k \mathbf{v}\|=|k|\|\mathbf{v}\| 1 answer 10 views If \mathbf{v} is a vector in R^{n}, and if k is any scalar, then prove that \|k \mathbf{v}\|=|k|\|\mathbf{v}\|If \mathbf{v} is a vector in R^{n}, and if k is any scalar, then prove that \|k \mathbf{v}\|=|k|\|\mathbf{v}\| ... close 1 answer 5 views close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Prove the orthogonal projection theorem 1 answer 4 views Prove the orthogonal projection theoremProve the orthogonal projection theorem ... close 1 answer 7 views If \mathbf{u} and \mathbf{v} are orthogonal vectors in R^{n} with the Euclidean inner product, then prove that$$ \|\mathbf{u}+\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2} $$If \mathbf{u} and \mathbf{v} are orthogonal vectors in R^{n} with the Euclidean inner product, then prove that$$ \|\mathbf{u}+\mathbf{v}\|^{2}= ...