For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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7
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118 views

Infer distance from a point to a line, from the distance from a point to a plane [Stewart P793 12.4.44]

I'm able to prove $44$, but how would one deduce $43$ from it without further industry, forthwith? $43$ seems like a reduced, 2D version of $44$? I'm not enquiring about individual proofs. $44.$ ...
6
votes
0answers
310 views

Finding the maximum number of subspaces of a vector space over finite field that satisfy these relations

I have a question and I am stuck. I was wondering if anyone has a thought, before I start a brute-force search. For $q$ a prime number and $n =6$, let $\mathbb {F}_{q}^{n}$ be an $n$-dimensional ...
5
votes
0answers
68 views

Decomposition of order-$n$ tensors

If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$. The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...
5
votes
0answers
73 views

When are all ring homomorphisms also algebra homomorphisms?

Let $k$ be an algebraically closed field, and let $A,B$ be two unitary $k$-algebras. In general, there are more ring homomorphisms $A\to B$ than there are $k$-algebra homomorphisms. More precisely, ...
5
votes
0answers
117 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
5
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0answers
61 views

Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...
5
votes
0answers
192 views

Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?

An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
4
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0answers
39 views

Can these characterisations of finite dimensionality be proven equivalent without using a basis?

I was wondering about how to define "finite dimensional" without talking about bases. Two possibilities occurred to me: Say $V$ is finite dimensional if the canonical inclusion $V\hookrightarrow ...
4
votes
0answers
87 views

Structure of a fuzzy subspace

Let $V$ be a vector space over a field $F$ and let $f$ be a function from $V$ to the interval $I:=[0,1]$ satisfying the condition that for any $a \in I$ the set $V_a:=\{v \in V | f(v) \ge a\}$ is a ...
4
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0answers
169 views

Vandermonde matrix

Let ${\bf G} \in\mathbb{C}^{M\times K}$ and ${\bf H} \in\mathbb{C}^{N\times K}$ are full-rank Vandermode matrices where $MN-1=K>N\geq N$, that is, ${\bf G}$ and ${\bf H}$ are fat. Let ${\bf F}= ...
3
votes
0answers
38 views

How to visualize cotangent spaces.

I was wondering how to intuitively and visually understand dual vector spaces and one-forms. So my question is (1), how to visualize cotangent spaces and (2), how to intuitively understand them? My ...
3
votes
0answers
36 views

Add vectors from a set to reach the goal vector, using the minimum possible cost

I am trying to solve a problem in an optimal way. The problem is as follows: We have an n-dimensional space In this space, we have a "finish" point with n coordinates, all non-negative We have a set ...
3
votes
0answers
33 views

Can we show it without involving that $V=V^{**}$ are canonically isomorph?

My text proves the following Theorem. Let $V$ be a vector space over $F$ and $B=\{ v_1, \ldots , v_n \}$ a basis of $V$. Then there is exactly one basis $B^*=\{ f_1, \ldots , f_n \}$ of $V^*$ with ...
3
votes
0answers
54 views

Find the projection of any vector onto the linear span and the normal from any vector to that span

Show that the vectors $u_1 = (1/9,4/9,8/9), u_2=(8/9,-4/9,1/9), u_3=(-4/9,-7/9,4/9)$ form an orthonormal basis of $\mathbb{R}^3$. Find the projection of any vector $x=(\xi_1,\xi_2,\xi_3) \in ...
3
votes
0answers
30 views

Proof check: any linear transformation can be represented as a matrix-vector product

I'm trying to prove that Theorem. Consider a linear transformation $T : \mathbb R^n \to \mathbb R^n$. The transformation $T$ can be represented as a matrix product $\mathbf x \mapsto A \mathbf ...
3
votes
0answers
224 views

Vector spaces - Multiplying by zero scalar yields zero vector

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space related axioms. ...
3
votes
0answers
48 views

Vector Space Verification

I just took an exam asking me if the following are a vector space over $\mathbb{R}$ assuming that the set of all real valued functions on the interval $[0,1]$ is a vector space with theoperations ...
3
votes
0answers
28 views

Higher Dimensional Right-Hand Rule

In seven dimensions, the cross product makes sense. Without resorting to nonvector tensors or exterior products (although they can be used to further explain), how does one perform this cross product ...
3
votes
0answers
52 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
3
votes
0answers
45 views

Find closest vector to a given vector from a particular set of vector

Let $x=\left(x_t\right)_{t=1}^n$ be a vector such that $$ x_t = \prod_{i=1}^t u_i, \tag{1} $$ where each parameters $u_i$ can take any of two value $$ u_i \in \left\{a,b \right\} = \left\{ 1.3, 0.8 ...
3
votes
0answers
48 views

Choosing an appropriate complete orthogonal basis

I have a function $f(x)$ which I want to represent as the sum over some complete orthogonal basis $\phi_i$ such that: $$ f(x) = \sum_{i} c_i \phi_i(x) $$ Where the $\phi_i$ are orthogonal with ...
3
votes
0answers
82 views

Basis for an infinite dimensional vector space.

Is there any good paper that focus on the topic of basis for infinite dimensional vector space that I can read/ study. I found some papers that mention about this topic online, but they are very brief ...
3
votes
0answers
75 views

Problem involving subspaces and linear transformations

I'm asking for some opinions about my proof! $V$ and $W$ are vector spaces, and $T : V \rightarrow W$ is a linear transformation. $Z$ is a subspace of $W$, and $U$ is the set of all $\textbf{x} \in ...
3
votes
0answers
291 views

How can I show that two infinite-dimensional vector spaces are isomorphic?

How can I show that two infinite-dimensional vector spaces are isomorphic? Can I define a function which maps a basis of the vector space $V$ to a basis of the vector space $W$? And are there more ...
3
votes
0answers
441 views

Show that if $S$ is a subspace of a vector space $V$, then $\dim (S) \leq \dim (V)$

Show that if $S$ is a subspace of a vector space $V$, then $\dim(S) \leq \dim(V)$. Furthermore, if $\dim(S)= \dim(V) < \infty$ thn $S=V$. Give an example to show that the finiteness is required ...
3
votes
0answers
212 views

How to show annihilator has dimension m-n (with Proof)

I would like to show the following: Given a vector spaces $V$, a subspace $S \subset V$ and an the dual space $V^*$ to $V$. Show that: $$\dim(N)+\dim(S) = \dim(V) = \dim(V^*)$$, where $N \subset V^*$ ...
3
votes
0answers
47 views

find the dimension of $W.$

Let $W=\{p(B):p \text{ is a polynomial with real coefficients}\},$ where $B=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}.$ Then find the dimension of $W.$ I have shown ...
3
votes
0answers
192 views

Divergence Theorem to prove equality of integrals

I'm trying to wrap my head around this problem - the interplay between $\nabla$ and $\Delta$ is doing my head in. It says to use the divergence theorem. Prove that $$\int_\Omega u \cdot \Delta v\, ...
3
votes
0answers
46 views

Symmetrizing a sequence of vectors

Given a finite set of real numbers $X_1, \ldots, X_n$, we can compute the first $n$ power sums of these numbers. From the power sums, the set $\{X_1, \ldots, X_n\}$ can be recovered. Essentially we ...
3
votes
0answers
787 views

Three-dimensional vectors and force systems

Full disclosure: this is a homework problem. However, I find myself stuck in the middle. The problem is below As shown, a system of cables suspends a crate weighing W = 350 . (Part C 1 figure) ...
3
votes
0answers
104 views

Self-absorbing subsets in a vector space

From planetmath Let $V$ be a vector space over a field $F$ equipped with a non-discrete valuation $|\cdot|:F\to \mathbb{R}$ . Let $A$ and $B$ be two subsets of $V$. Then $A$ is said to absorb ...
2
votes
0answers
6 views

Mean value theorem and scalar field proof

Assume that f′(x;y)=0 for every x in some n-ball B(a) and for every vector y. Use the mean value theorem to prove that f is constant on B(a). And if f′(x;y)=0 for a fixed vector y and for every x in ...
2
votes
0answers
29 views

Prove that $U_1\cup U_2$ is a subspace of $V$ $\iff$ $U_1\subseteq U_2$ or $U_2\subseteq U_1$ $\triangle$

Let $V$ be a vector space over some field. Let $U_1$ be a subspace of $V$. Let $U_2$ be a subspace of $V$. Prove that $U_1\cup U_2$ is a subspace of $V$ is equivalent to $U_1\subseteq U_2$ or ...
2
votes
0answers
24 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
2
votes
0answers
38 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.9, Problem 12

If $f_1, \ldots, f_p$ are linear functionals on an $n$-dimensional vector space $X$, where $p<n$, then how to show that there is a vector $x \ne 0$ in $X$ such that $f_1(x) = 0, \ldots, f_p(x)=0$? ...
2
votes
0answers
82 views

$\rm span(S_1) + \rm span(S_2) = \rm span(S_1 \cup S_2)$ for infinite sets

I have these two definitions of span: Span: Suppose a vector space $(V,+,\cdot)$, and $$S = \{u_1,\cdots,u_n\}$$ (and $S$ is a subset of $V$, not a subspace) ...
2
votes
0answers
44 views

Dimension of the set of mxn complex component matrices over real numbers

What is the dimension of $M_m$$_x$$_n$$\mathbb{C}$ when considered as a vector space over $\mathbb{R}$? My approach: If I take an mxn matrix of complex entries and I want to write this as a linear ...
2
votes
0answers
25 views

Problem 1.6 in Edwards multivariable calculus

The problem requires a proof that if $S$ is a set and $\mathscr{F}(S, \mathbb{R})$ is the set of all functions $S \to \mathbb{R}$, then $\mathscr{F}(S, \mathbb{R})$ is a vector space. Let $f, g: S ...
2
votes
0answers
17 views

Set of polynomials a subspace of P3?

Is the set of all polynomials of the form a0+a1x, where a0 and a1 are real numbers a subspace of P3? My book says it is not. Both closure under addition and scalar multiplication hold, so I don't ...
2
votes
0answers
21 views

How to understand affine space and affine transformation

As far as I know, the affine space is a space without origin point. Some others define affine space as $$A=\{\sum_{i=1}^N \alpha_i \boldsymbol{v_i|\sum_{i=1}^N}\alpha_i=1\}$$ How do we relate these ...
2
votes
0answers
33 views

Norms on $\mathbb{R}$ seen as a $\mathbb{Q}$-vector space.

this is not really a question : I had some ideas on topics I don't feel secure with. I expose these hereafter : are there any mistakes in my reasonning ? Also, if anyone knows a good read about this ...
2
votes
0answers
18 views

Linear algebra (Coordinates)

Question: Find the coordinates of $x=(1,0,0)$ in relation to base $$B=\{(1,1,1),(-1,1,0),(1,0,-1)\}.$$ I tried: $a,b,c\in R$ such that $$a(1,1,1)+b(-1,1,0)+c(1,0,-1)=(1,0,0)=x$$ but I'm not sure ...
2
votes
0answers
38 views

Is a vector space over $\mathbb{C}$ also a vector space over $\mathbb{R}$?

Let $V = \{(a_1, a_2,\ldots, a_n):a_i$ is an element of $\mathbb{C}$ for $i = 1,2,\ldots, n\}$; so $V$ is a vector space over $\mathbb{C}$. Is $V$ a vector space over the field of real numbers with ...
2
votes
0answers
185 views

Vector space basis change: is this “index-free” notation correct?

There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ...
2
votes
0answers
28 views

Blocking set for cosets of codimension $2$

In this paper following theorem is proved: If $V$ is vector space of dimension $n$ over a finite field $F$ of $q$ elements then any subset of $V$ which meets every hyperplane of $V$ contains at least ...
2
votes
0answers
41 views

How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
2
votes
0answers
31 views

Notation for a vector space: $(\mathbb{C}^\infty)^{\otimes L}$

In a paper, the authors use the notation $(\mathbb{C}^\infty)^{\otimes L}$, where $L$ is a constant, for a vector space, but they do not give a definition. They also implicitly introduce an inner ...
2
votes
0answers
30 views

About an orthogonal complement theorem

Let $W$ be a subspace of $\mathbb{R}^n$. For any vector $x \in \mathbb{R}^n$, there will one unique vector $y \in W$ that fulfils: $$(x-y) \perp w \ \ : \ \ \forall w \in W$$ I have trouble ...
2
votes
0answers
22 views

Vectorial product analog operation in 4+ dimensions?

I am thinking about a such operation. Which it need to have: It needs to be $\mathbb{R}^n\times{\mathbb{R}^n}\rightarrow\mathbb{R}^n$ The result needs to be perpendicular to the arguments (thus, ...
2
votes
0answers
66 views

kernel space of linear combination of matrices

Suppose $A$ and $B$ are $N\times N$ matrices so that for every $x$ and $y$, $xA+yB$ has a kernel of dimension at least $2$. Is it necessarily true that $\ker(A)\cap\ker(B)$ has dimension at least ...