# Recent questions tagged real

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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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A $3 \times 3$ real matrix need not have any real eigenvalues.
A $3 \times 3$ real matrix need not have any real eigenvalues.True or False - and Why? a) A $3 \times 3$ real matrix need not have any real eigenvalues. b) If an $n \times n$ matrix $A$ is invertible, then ...
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For non-zero real numbers one uses $\frac{1}{a}-\frac{1}{b}=\frac{b-a}{a b}$. Verify the following analog for invertible matrices $A, B$ :
For non-zero real numbers one uses $\frac{1}{a}-\frac{1}{b}=\frac{b-a}{a b}$. Verify the following analog for invertible matrices $A, B$ :For non-zero real numbers one uses $\frac{1}{a}-\frac{1}{b}=\frac{b-a}{a b}$. Verify the following analog for invertible matrices $A, B$ : \ A^{- ...
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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 Let \(L: V \rightarrow V$ be a linear map on a vector space $V$.
Let $L: V \rightarrow V$ be a linear map on a vector space $V$.Let $L: V \rightarrow V$ be a linear map on a vector space $V$. a) Show that $\operatorname{ker} L \subset \operatorname{ker} L^{2}$ and, more g ...
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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 $A$ be a real matrix, not necessarily square.
Let $A$ be a real matrix, not necessarily square.Let $A$ be a real matrix, not necessarily square. &nbsp; a) If $A$ is onto, show that $A^{}$ is one-to-one. b) If $A$ is one-to-one, show ...
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Let $\mathcal{S}$ be the linear space of infinite sequences of real numbers $x:=\left(x_{1}, x_{2}, \ldots\right) .$ Define the linear map $L: \mathcal{S} \rightarrow \mathcal{S}$ byLet $\mathcal{S}$ be the linear space of infinite sequences of real numbers $x:=\left(x_{1}, x_{2}, \ldots\right) .$ Define the linear map $L: \m ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find the dimension of \(A$ considered as a real vector space.
Find the dimension of $A$ considered as a real vector space.Consider the two linear transformations on the vector space $V=\mathbf{R}^{n}$ : $R=$ right shift: $\left(x_{1}, \ldots, x_{n}\right) \rightarrow ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that \(\mathcal{L}$ and $\mathcal{R}$ are linear spaces and compute their dimensions.
Show that $\mathcal{L}$ and $\mathcal{R}$ are linear spaces and compute their dimensions.Say $A \in M(n, \mathbb{F})$ has rank $k$. Define \ \mathcal{L}:=\{B \in M(n, \mathbb{F}) \mid B A=0\} \quad \text { and } \quad \mathcal{R}:=\{C ...
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Let $A$ be a square real matrix all of whose eigenvalues are zero. Show that $A$ is diagonalizable (that is, similar to a possibly comples diagonal matrix) if and only if $A=0$.Let $A$ be a square real matrix all of whose eigenvalues are zero. Show that $A$ is diagonalizable (that is, similar to a possibly comples diagona ...
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Give an example of a square real matrix that has rank 2 and all of whose eigenvalues are zero.
Give an example of a square real matrix that has rank 2 and all of whose eigenvalues are zero.Give an example of a square real matrix that has rank 2 and all of whose eigenvalues are zero. ...
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Let $V, W$ be two-dimensional real vector spaces, and let $f_{1}, \ldots, f_{5}$ be linear transformations from $V$ to $W$.
Let $V, W$ be two-dimensional real vector spaces, and let $f_{1}, \ldots, f_{5}$ be linear transformations from $V$ to $W$.Let $V, W$ be two-dimensional real vector spaces, and let $f_{1}, \ldots, f_{5}$ be linear transformations from $V$ to $W$. Show that there ex ...
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Let $\mathcal{M}_{(3,2)}$ be the linear space of all $3 \times 2$ real matrices and let the linear map $L$ : $\mathcal{M}_{(3,2)} \rightarrow \mathbb{R}^{5}$ be onto. Compute the dimension of the nullspace of $L$.Let $\mathcal{M}_{(3,2)}$ be the linear space of all $3 \times 2$ real matrices and let the linear map $L$ : $\mathcal{M}_{(3,2)} \rightarrow \ ... 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 real (or complex) matrix. Then $A$ is invertible if and only if zero is not an eigenvalue. Proof or counterexample.
Let $A$ be a square real (or complex) matrix. Then $A$ is invertible if and only if zero is not an eigenvalue. Proof or counterexample.Let $A$ be a square real (or complex) matrix. Then $A$ is invertible if and only if zero is not an eigenvalue. Proof or counterexample. ...
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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 Find a real \(2 \times 2$ matrix $A$ (other than $A=I$ ) such that $A^{5}=I$.
Find a real $2 \times 2$ matrix $A$ (other than $A=I$ ) such that $A^{5}=I$.Find a real $2 \times 2$ matrix $A$ (other than $A=I$ ) such that $A^{5}=I$. ...
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Let $U$ and $V$ both be two-dimensional subspaces of $\mathbb{R}^{5}$, and let $W=U \cap V$. Find all possible values for the dimension of $W$.
Let $U$ and $V$ both be two-dimensional subspaces of $\mathbb{R}^{5}$, and let $W=U \cap V$. Find all possible values for the dimension of $W$.Let $U$ and $V$ both be two-dimensional subspaces of $\mathbb{R}^{5}$, and let $W=U \cap V$. Find all possible values for the dimension of $W ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(C(\mathbb{R})$ be the linear space of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$.
Let $C(\mathbb{R})$ be the linear space of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$.Let $C(\mathbb{R})$ be the linear space of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$. a) Let $S_{c}$ be the set of different ...
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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 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 ...
Which of the following sets are linear spaces?Which of the following sets are linear spaces? a) $\left\{X=\left(x_{1}, x_{2}, x_{3}\right)\right.$ in $\mathbb{R}^{3}$ with the property $\lef ... 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 ...