Questions tagged [vector-spaces]
For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars
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Is there a counter example to disprove the following regarding vector addition in binary field?
Let $\{\mathbf{a}_1 , \mathbf{a}_2 , \mathbf{a}_3 , ...., \mathbf{a}_{30}\}\subset \mathbb{F}_2^{15}$ denote the set of binary vectors. Define the set of integers $\{p_k\}_{k=1}^{14}$ as$3 \leq p_1 &...
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The $8$ laws of the definition of vector spaces are not independent. What is the relationship among them? [duplicate]
I am learning linear algebra, and have a question about the eight laws in the definition of a vector space.
A non-empty set $V$ is a vector space on field $F$, if addition $+:V\times V\to V$ and ...
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Questions about how to show $d_1+\cdots +d_n-n+1 \leq {\text{dim}}_k k[x_1,\ldots,x_n]/\mathfrak{a}\leq d_1d_2\cdots d_n\quad $
The following are from Froberg's "Introduction to Grobner bases" , and Hungerford's undergraduate "Abstract Algebra" text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\...
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What is the connection between bilinear and quadratic forms.
I know that a bilinear form $B$ on the $\mathbb R$-vector space $\mathbb R^n$ is defined to be a map $B:\mathbb R^n\times \mathbb R^n\to \mathbb R$ which is linear in each coordinate.We know that a ...
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Showing that $k[x_1,\ldots,x_n]/\mathfrak{a}$ is a finite dimensional vector space over $k$ assuming basic linear algebra and min amount of abs alg.
The following are from Froberg's Introduction to Grobner bases, and Hungerford's undergraduate Abstract Algebra text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\backsimeq (k[x_1,\ldots,x_{n-1}])...
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Applying vector decomposition multiple times and RH orthonormal bases
I want to show that I can write any 3D vector $v$ in components with respect to the right handed orthonormal basis $\{e_1, e_2, e_3\}$ (i.e. three perpendicular unit vectors $\{e_1, e_2, e_3\}$ such ...
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Constructing the interval [0, 1) via inverse powers of 2
If $x$ is rational and in the interval ${[0,1)}$, is it always possible to find constants $a_1, a_2, ..., a_n\in\{-1, 0, 1\}$ such that for some integer $n\geq{1}$, $x = a_1\cdot2^{-1} + a_2\cdot{2^{-...
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Is this a valid vector space? (Question Verification) [closed]
Hi - please see the question above. I have a problem with proving associativity i.e. that $(\alpha \beta)v = \alpha(\beta v)$ where $\alpha, \beta \in \mathbb{R}$ and $v \in \mathbb{R}_+$.
I think ...
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Distance between subspaces after premultiplication by diagonal matrice
Let $\mathcal{E}_1$ and $\mathcal{E}_2$ be two k dimensional subspaces in $\mathbb{R}^n$ and two $n\times K$ matrices $E_1$,$E_2$ are basis matrix. Distance between two subspaces is defined to be the ...
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Orthogonal complement with respect to a subspace, and then with respect to the larger space.
Suppose I have the subspaces $W\leq V \leq \mathbb{F}_q^n$, with $n$ finite. Let $\langle ,\rangle\colon\mathbb{F}_q^n\times \mathbb{F}_q^n \rightarrow \mathbb{F}_q$ be the dot-product. If I then take ...
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Bilinear forms and reflexivity [duplicate]
Let $V$ be a finite dimensional vector space over a field $\mathbb{K}$, and let $\varphi: V \times V \to \mathbb{K}$ be a bilinear form on $V$. Let's give some definitions. We say that $\varphi$ is ...
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Tensor Product of a vector space with itself.
I've been reading up on tensor products and have been coming up blank on how to think about $V \otimes_F V$, where $V$ is a vector space over a field $F$.
I only care about what is happening when $\...
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Little trouble understanding a uniqueness proof?
I'm reading Postnikov's Analytic Geometry. Here:
What is happening in there? I understand they may be somehow showing that $x$ is unique but I don't understand what is happening in the equations.
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Does $M_{2,3}(\mathbb{R})$ define a vector space adequately?
I was reading my instructor's notes on vector spaces. I came across the following:
Example 9.2.2 Let $M_{2,3}(\mathbb{R})$ be the set of all $2\times3$ matrices over $\mathbb R$. Show that $M_{2,3}(\...
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Orthonormal basis for $\mathbb{C}^2$ over $\mathbb{R}$ [closed]
$\mathbb{C}^2$ is a 4-dimensional vector space over $\mathbb{R}$ with basis $\left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, ...
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