# Recent questions tagged space

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Let $A$ be a positive definite $n \times n$ real matrix, $\vec{b}$ a real vector, and $\vec{N}$ a real unit vector.
Let $A$ be a positive definite $n \times n$ real matrix, $\vec{b}$ a real vector, and $\vec{N}$ a real unit vector.Let $A$ be a positive definite $n \times n$ real matrix, $\vec{b}$ a real vector, and $\vec{N}$ a real unit vector. a) For which value(s) of t ...
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Let $\vec{x}$ and $\vec{p}$ be points in $\mathbb{R}^{n}$. Under what conditions on the scalar $c$ is the set $\|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0$
Let $\vec{x}$ and $\vec{p}$ be points in $\mathbb{R}^{n}$. Under what conditions on the scalar $c$ is the set $\|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0$a) Let $\vec{x}$ and $\vec{p}$ be points in $\mathbb{R}^{n}$. Under what conditions on the scalar $c$ is the set \ \|\vec{x}\|^{2}+2\langle\v ...
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Let $V \subset \mathbb{R}^{n}$ be a linear space, $Q: R^{n} \rightarrow V^{\perp}$ the orthogonal projection into $V^{\perp}$, and $x \in \mathbb{R}^{n}$ a given vector.Dual variational problems Let $V \subset \mathbb{R}^{n}$ be a linear space, $Q: R^{n} \rightarrow V^{\perp}$ the orthogonal projection into $V^ ... close 0 answers 11 views Let \(C[-1,1]$ be the real inner product space consisting of all continuous functions $f:[-1,1] \rightarrow \mathbb{R}$, with the inner product $\langle f, g\rangle:=\int_{-1}^{1} f(x) g(x) d x$.Let $C-1,1$ be the real inner product space consisting of all continuous functions $f:-1,1 \rightarrow \mathbb{R}$, with the inner product $\ ... 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 $p(x)=a+b x+c x^{2}$ of degree at most 2 with the inner product $\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x$.
Let $\mathcal{P}_{2}$ be the space of polynomials $p(x)=a+b x+c x^{2}$ of degree at most 2 with the inner product $\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x$.Let $\mathcal{P}_{2}$ be the space of polynomials $p(x)=a+b x+c x^{2}$ of degree at most 2 with the inner product $\langle p, q\rangle=\int_{-1}^ ... 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 quadratic polynomials.
Let $\mathcal{P}_{2}$ be the space of quadratic polynomials.Let $\mathcal{P}_{2}$ be the space of quadratic polynomials. a) Show that $\langle f, g\rangle=f(-1) g(-1)+f(0) g(0)+f(1) g(1)$ is an inner produ ...
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In a complex vector space (with a hermitian inner product), if a matrix $A$ satisfies $\langle X, A X\rangle=0$ for all vectors $X$, show that $A=0$. [The previous problem shows that this is false in a real vector space].In a complex vector space (with a hermitian inner product), if a matrix $A$ satisfies $\langle X, A X\rangle=0$ for all vectors $X$, show that \ ...
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Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map defined by the matrix $A$.
Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map defined by the matrix $A$.Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map defined by the matrix $A$. If the matrix $B$ satisfies the relation $\langle ... close 0 answers 6 views Let \(A$ be a positive definite $n \times n$ real matrix, $b \in \mathbb{R}^{n}$, and consider the quadratic polynomial $Q(x):=\frac{1}{2}\langle x, A x\rangle-\langle b, x\rangle$Let $A$ be a positive definite $n \times n$ real matrix, $b \in \mathbb{R}^{n}$, and consider the quadratic polynomial \ Q(x):=\frac{1}{2}\lang ...
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Let $\ell$ be any linear functional. Show there is a unique vector $v \in \mathbb{R}^{n}$ so that $\ell(x):=\langle x, v\rangle$.
Let $\ell$ be any linear functional. Show there is a unique vector $v \in \mathbb{R}^{n}$ so that $\ell(x):=\langle x, v\rangle$.LINEAR FUNCTIONALS In $R^{n}$ with the usual inner product, a linear functional $\ell:$ $\mathbb{R}^{n} \rightarrow \mathbb{R}$ is just a line ...
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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 $V, W$ 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 $\vec{e}_{1}, \vec{e}_{2}$, and $\vec{e}_{3}$ be the standard basis for $\mathbb{R}^{3}$ and let $L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a linear transformation with the propertiesLet $\vec{e}_{1}, \vec{e}_{2}$, and $\vec{e}_{3}$ be the standard basis for $\mathbb{R}^{3}$ and let $L: \mathbb{R}^{3} \rightarrow \mathbb{R}^ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(U \subset V$ and $W$ be finite dimensional linear spaces and $L: V \rightarrow W$ a linear map. Show that
Let $U \subset V$ and $W$ be finite dimensional linear spaces and $L: V \rightarrow W$ a linear map. Show thatLet $U \subset V$ and $W$ be finite dimensional linear spaces and $L: V \rightarrow W$ a linear map. Show that \ \operatorname{dim}\left(\left. ...
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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 $B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}$.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 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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Which of the following are not a basis for the vector space of all symmetric $2 \times 2$ matrices? Why?
Which of the following are not a basis for the vector space of all symmetric $2 \times 2$ matrices? Why?Which of the following are not a basis for the vector space of all symmetric $2 \times 2$ matrices? Why? &nbsp; a) $\left(\begin{array}{ll}1 &amp ... close 0 answers 14 views 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 Give an example of a linear transformation \(L: V \rightarrow V$ (or show that there is no such transformation) for which:
Give an example of a linear transformation $L: V \rightarrow V$ (or show that there is no such transformation) for which:Let $V$ be a vector space with $\operatorname{dim} V=10$ and let $L: V \rightarrow V$ be a linear transformation. Consider $L^{k}: V \rightarro ... close 0 answers 11 views Given a unit vector \(\mathbf{w} \in \mathbb{R}^{n}$, let $W=\operatorname{span}\{\mathbf{w}\}$ and consider the linear map $T: \mathbb{R}^{n} \rightarrow$ $\mathbb{R}^{n}$ defined byGiven a unit vector $\mathbf{w} \in \mathbb{R}^{n}$, let $W=\operatorname{span}\{\mathbf{w}\}$ and consider the linear map $T: \mathbb{R}^{n} \ri ... 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 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 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 ...