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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8
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190 views

A conjecture about vector space

Let $V$ be a $(r+1)$-dimensional vector space, and $p$ be a positive integer and $1\leq p\leq r-1$. Let $$X=\{v_1,\cdots,v_{2r+1-p}\}\subseteq V$$ be a finite set containing $(2r+1-p)$ different ...
6
votes
0answers
96 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
227 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
86 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
votes
0answers
203 views

Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
5
votes
0answers
37 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
169 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
votes
0answers
88 views

Number of zeros equal number of linearly independent analytic functions

I'm trying to read this paper and I'm stuck on a particular point. The authors are constructing an analytic function $f(z)$ which have to satisfy the following boundary conditions: ...
4
votes
0answers
78 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
votes
0answers
142 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
32 views

Given three points in $\mathbb R^3$ that define a plane. Need to find the normal of the plane.

I came across this question and it has been troubling me for a while... A plane in $\mathbb R^3$ that contains three points is defined as $A=(1, 2, 3)$, $B=(0, 1, 4)$, $C=(2, 1, -7)$... I have to find ...
3
votes
0answers
17 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
45 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
33 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
40 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
53 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
66 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
206 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
44 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
164 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
41 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
638 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
99 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
25 views

Strict subsets in vector space

Find proper subsets $S_1,S_2,S_3 \neq \{(0,0)\}$ in vector space $\mathbb{R}^2$ so that: $$S_1 + S_2 \subsetneq S_1$$ $$S_2 \subsetneq S_2+S_2$$ $$S_3+S_3=S_3$$ I wrote out the addition definitions: ...
2
votes
0answers
30 views

Basis for the power vector space of a vector space

Let $V$ be a vector space over a field $F$ , let $P(V)$ denote the power set of $V$ ; for $A, B \in P(V) $ and $ a \in F$ , define $A +' B :=${$x+y : x\in A , y\in B$ } and $a.A:=${$ax : x\in A$ } , ...
2
votes
0answers
48 views

Real and complex vector spaces

Suppose that $V$ is a real finite-dimensional vector space and let $V_\mathbb{C}=V\otimes_{\mathbb{R}}\mathbb{C}$ be its complexification. Now let $W\subset V_\mathbb{C}$ be a complex subspace. ...
2
votes
0answers
66 views

Isomorphism,on ${R}^4$

I dont understand what the function is for part (a) such that a mapping from $X\in T_p{R}^4$ to $w(X,-)\in T^{\star}_p{R}^4$ be an isomorfism!. So Consider on ${R}^4=(x_1,y_1,x_2,y_2)$ the ...
2
votes
0answers
33 views

The definition of the exchange lemma

We've just learnt about the exchange lemma, and I was looking at someone else's notes - I don't understand the point 2. How can $\psi_t \in \{\psi_1,...,\psi_n \}$? Should it not be in ...
2
votes
0answers
60 views

Linear Operators: Continuous $\Rightarrow$ Bounded

Let $T:V\rightarrow V'$ be a continuous linear operator between two normed vector spaces $V,V'$. Show that it is bounded. Continuity is defined as $\lim_{n}\|x_n-x\|=0\Rightarrow ...
2
votes
0answers
73 views

Find all the invariant subspaces of T

T is a linear transformation, defined as the following: $T(p(x)) = xp(x)$, $T\colon R[X]\to R[X]$ Find all the invariant subspaces of $T$. As I see it, only the trivial subspaces $0$, $R[X]$ are ...
2
votes
0answers
71 views

Recognizing pure tensors in tensor product of vector spaces

Let $V$ be a vector space and let $\{e_i\}$ be a basis for it. Then $\{e_I\equiv e_{i_1}\otimes...\otimes e_{i_r}\}$ is a basis for $V\otimes ... \otimes V$. Suppose I am given an element $w=\sum a_I ...
2
votes
0answers
21 views

A large set of low dimensional vectors in $\mathbb{F}_2^L$, which sums of any small subset do not cancel.

Fix a number $n$ and $L=O(\log n)$. Let $S=\{v_1,\dots,v_n\}$ be a set of vectors where $v_i\in \mathbb{F}_2^L$. We say that $S$ is "$\alpha$-good" iff for any nonempty subset $T \subset S$ where ...
2
votes
0answers
35 views

Indecomposable vector space

Let us define $V$ as a quotient space $\mathbb{K}[t]/(p^m)$, where $p$ is an irreducible polynomial. Condsider the linear operator $\phi\in Hom_{\mathbb{K}}(V,V)$, which sends each $q+(p^m)$ to ...
2
votes
0answers
62 views

Prove that if W is a subspace of a vector space V

Prove that if $W$ is a subspace of a vector space $V$ and $w_1, w_2, ..., w_n$ are in $W$, then $a_1w_1 + a_2w_2 + ... + a_nw_n \in W$ for any scalars $a_1, a_2, ..., a_n$. My solution is we have ...
2
votes
0answers
174 views

Horizontal and vertical tangent space of Orthogonal group

We know for the orthogonal group n-by-n orthogonal matrices, the tangents are given by $X^T\Delta + \Delta^TX = 0$ where $\Delta$ is the tangent. Now I was reading about the vertical and horizontal ...
2
votes
0answers
124 views

Inner product of two complex vectors?

Given $A \in \mathbb R^{m \times n}$real matrix. If $\langle x,y\rangle =y^{*}x$ for all $x,y\in \mathbb C^{n \times 1}$, can someone help me find the relationship between the following two ...
2
votes
0answers
271 views

Sum of dimensions of subspaces, dimension of sum of subspaces

I am kind of confused when it comes to subspaces. I have difficulty grasping them conceptually. In one of my questions there is the following: Find two subspaces $H$ and $K$ of $\mathbb{R}^3$ such ...
2
votes
0answers
58 views

Difficulty in understanding a sentence from a paper

During reading a paper I faced following sentence: Mathematically, different one dimensional real vector spaces can represent different basic dimensions. (basic dimensions, like time, length, ...
2
votes
0answers
56 views

Spanning Hadamard product powers (Schur products)

Fix two vectors $\mathbf u$ and $\mathbf v$ in $\mathbb R^k$, and let $\circ$ denote the coordinate-wise Hadamard / Schur product, i.e. $\mathbf u\circ\mathbf v$ has coordinates $w_i=u_iv_i$. Write ...
2
votes
0answers
277 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 ...
2
votes
0answers
75 views

Set of all odd complex polynomials - complex vector space

Is the set of all odd complex polynomials a complex vector space? I'm given the following definition of a vector space: A vector space $V$ over the field $\mathbb F$ is a set $V$ of vectors, a field ...
2
votes
0answers
162 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 $N \subset V^*$ ...
2
votes
0answers
92 views

Transposition of Composition is Reversed Composition of Transpositions

I'm trying to show that $(UT)^*=T^*U^*$. Here is my effort: Consider the following data: \begin{array}{lcl} T:V\rightarrow W & \leadsto & T^*:W^*\rightarrow V^* \\ U:W\rightarrow Z & ...
2
votes
0answers
43 views

How prove this $\frac{|x-z|}{|x-y|}=1+\frac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$

prove that $$\dfrac{|x-z|}{|x-y|}=1+\dfrac{1}{|x|}\hat{x}\cdot(y-z)+O(1/|x|^2)$$ for $|x|\longrightarrow \infty$ where $$\hat{x}=\dfrac{x}{|x|}$$ This problem from book,following is my idea: ...
2
votes
0answers
60 views

Is there a name for this type of matrix?

I am preparing to publish an academic article on computational efficiency and image processing. In my work, I have come across what I can best describe as a non-square skew (symmetric or repeating) ...
2
votes
0answers
85 views

Simple linear operator?

From Wikipedia a linear operator T on a finite-dimensional vector space is semi-simple if every T-invariant subspace has a complementary T-invariant subspace. I wonder if there is a concept for ...
2
votes
0answers
48 views

Isomorphism of $P(V)$ and $P(V^*)$

Let $V$ be a finite-dimensional left vector space over a division ring $K$, and let $V^*$ the dual right vector space (consisting of all linear functions from $V$ to $K$). We can (and will) treat ...
2
votes
0answers
51 views

Continuity of linear form

Let $E=\mathbb{R}[X]$ We define $N:\, P \to \sum_{n=0}^{\infty} { |P^{(n)}(n)|}$ ($P^{(n)}$ being the $n$-th derivative) , it is not hard to prove that $N$ is a norm on $E$. Help me to study the ...
2
votes
0answers
44 views

Randomized Solution to a System of Inequalities

Given a set of $\mathbf v_i \in \{0,1\}^k$ for $i=1,\dots,n$ and a vector $\mathbf x \in [0,1]^k$, we want to decide if the following inequality holds or not: $$ \mathbf x \le \sum_{i=1}^n \alpha_i ...
2
votes
0answers
101 views

Is the notion of a dual space related to the set of polynomial functions on an affine algebraic variety?

Let $M$ be an affine algebraic variety and consider the ring of polynomial functions on $M$, $\mathcal{O}(M):=\{f: M\to k : f\text{ a polynomial}\}$. If $k$ is algebraically closed we can recover our ...