Recent questions tagged proof

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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 $A$ be a square matrix. In the following, a sequence of matrices $C_{j}$ converges if all of its elements converge.
Let $A$ be a square matrix. In the following, a sequence of matrices $C_{j}$ converges if all of its elements converge.Let $A$ be a square matrix. In the following, a sequence of matrices $C_{j}$ converges if all of its elements converge. Prove that the following a ...
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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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Show that $p(\lambda)=\lambda^{n}+a_{n-1} \lambda^{n-1}+\cdots+a_{0}$ is the characteristic polynomial of the matrix
Show that $p(\lambda)=\lambda^{n}+a_{n-1} \lambda^{n-1}+\cdots+a_{0}$ is the characteristic polynomial of the matrixShow that $p(\lambda)=\lambda^{n}+a_{n-1} \lambda^{n-1}+\cdots+a_{0}$ is the characteristic polynomial of the matrix \ A=\left(\begin{array}{ccccc} ...
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The characteristic polynomial of a square matrix is the polynomial $p(\lambda)=\operatorname{det}(\lambda I-$ $A$ ).
The characteristic polynomial of a square matrix is the polynomial $p(\lambda)=\operatorname{det}(\lambda I-$ $A$ ).The characteristic polynomial of a square matrix is the polynomial $p(\lambda)=\operatorname{det}(\lambda I-$ $A$ ). a) If two square matrices ar ...
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Let $C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions:
Let $C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions:Let $C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions: a) $\operatorname{det}\left ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A$ be a square matrix. Proof or Counterexample. If $A$ is diagonalizable, then so is $A^{2}$.
Let $A$ be a square matrix. Proof or Counterexample. If $A$ is diagonalizable, then so is $A^{2}$.Let $A$ be a square matrix. Proof or Counterexample. a) If $A$ is diagonalizable, then so is $A^{2}$. b) If $A^{2}$ is diagonalizable, then so ...
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There are no square matrices $A, B$ with the property that $A B-B A=I .$ Proof or counterexample.
There are no square matrices $A, B$ with the property that $A B-B A=I .$ Proof or counterexample.There are no square matrices $A, B$ with the property that $A B-B A=I .$ Proof or counterexample. REMARK: In quantum physics, the operators $A u ... 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 $A=\left(a_{i j}\right)$ be an $n \times n$ matrix whose rank is 1 . Let $v:=\left(v_{1}, \ldots, v_{n}\right) \neq 0$ be a basis for the image of $A$.
Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix whose rank is 1 . Let $v:=\left(v_{1}, \ldots, v_{n}\right) \neq 0$ be a basis for the image of $A$.Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix whose rank is 1 . Let $v:=\left(v_{1}, \ldots, v_{n}\right) \neq 0$ be a basis for the im ...
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Let $A$ and $B$ be $n \times n$ matrices with the property that $A B=0 .$ For each of the following give a proof or counterexample.
Let $A$ and $B$ be $n \times n$ matrices with the property that $A B=0 .$ For each of the following give a proof or counterexample.Let $A$ and $B$ be $n \times n$ matrices with the property that $A B=0 .$ For each of the following give a proof or counterexample. a) Every ...
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Let $L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample.
Let $L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample.Let $L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample. a) If $L^{2}=0$ then $L=0$. b) If $L^{2}=L$ the ...
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Let $A, B$, and $C$ be $n \times n$ matrices.
Let $A, B$, and $C$ be $n \times n$ matrices.Let $A, B$, and $C$ be $n \times n$ matrices. a) If $A^{2}$ is invertible, show that $A$ is invertible. NOTE: You cannot naively use the f ...
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Let $A$ be a real square matrix satisfying $A^{17}=0$.
Let $A$ be a real square matrix satisfying $A^{17}=0$.Let $A$ be a real square matrix satisfying $A^{17}=0$. a) Show that the matrix $I-A$ is invertible. b) If $B$ is an invertible matrix, is $B ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Proof or counterexample. In these \(L$ is a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$, so its representation will be as a $2 \times 2$ matrix.
Proof or counterexample. In these $L$ is a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$, so its representation will be as a $2 \times 2$ matrix.Proof or counterexample. In these $L$ is a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$, so its representation will be as a $2 \times 2 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A$ be a square matrix with integer elements. For each of the following give a proof or counterexample.
Let $A$ be a square matrix with integer elements. For each of the following give a proof or counterexample.Let $A$ be a square matrix with integer elements. For each of the following give a proof or counterexample. a) If $\operatorname{det}(A)=\pm 1$, t ...
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Let $A$ and $B$ be $n \times n$ matrices with $A B=0$. Give a proof or counterexample for each of the following.
Let $A$ and $B$ be $n \times n$ matrices with $A B=0$. Give a proof or counterexample for each of the following.Let $A$ and $B$ be $n \times n$ matrices with $A B=0$. Give a proof or counterexample for each of the following. &nbsp; a) Either $A=0$ or ...
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What are the components of mathematical discourse?
What are the components of mathematical discourse?What are the components of mathematical discourse? ...
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What is mathematical discourse?
What is mathematical discourse?What is mathematical discourse? ...
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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 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 ...
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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 $\|k \mathbf{v}\|=|k|\|\mathbf{v}\|$If $\mathbf{v}$ is a vector in $R^{n}$, and if $k$ is any scalar, then prove that $\|k \mathbf{v}\|=|k|\|\mathbf{v}\|$ ...
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Match the reasons to the lines of the following proof.
Match the reasons to the lines of the following proof. Match the reasons to the lines of the following proof. Let $T$ be a triangle with vertices $u, v$, and $w$ in $\mathrm{C}$ where the vertices are n ...
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You may choose the same option for matching more than once. Read through the following proof and match the lines with the reasons given below.
You may choose the same option for matching more than once. Read through the following proof and spot an error or not in the lines $L 1$ to $L 5$.
You may choose the same option for matching more than once. Read through the following proof and spot an error or not in the lines $L 1$ to $L 5$. You may choose the same option for matching more than once. Read through the following proof and spot an error or not in the lines $L 1$ to $L 5$. ...