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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110 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
268 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
65 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
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0answers
105 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
46 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
184 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
81 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
156 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
23 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
89 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
86 views

How to prove that this set is closed.

Lets say $a_1, a_n$ are normed vectors. Why is the set $C = \{\Sigma_{i=1}^n \lambda_ia_i: \lambda_i \ge0\}$ closed? The $\lambda$'s can be any non-negative numbers. So C is the set of all ...
3
votes
0answers
41 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
39 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
23 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
49 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
41 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
43 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
70 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
68 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
240 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
361 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
45 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
180 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
42 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
724 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
100 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
17 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
31 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
24 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
35 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
26 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
26 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
59 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 ...
2
votes
0answers
101 views

A basis for the column space of a real matrix

Let $A$ be a real square matrix, and let its column space be $$\mathrm{col}(A)=\{y\in\mathbb{C}^n:y=Ax\text{ for some } x\in\mathbb{C}^n\}.$$ Under what conditions is $\mathrm{col}(A)$ spanned by ...
2
votes
0answers
27 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
36 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
87 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
67 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
41 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
84 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
38 views

Fine-grained way to measure infinity

It is known that the cardinality of $R$ is equal to the cardinality of $R^2$, $R^3$, etc. But, intuitively these sets have different sizes. A possible way to formalize this intuition is to talk about ...
2
votes
0answers
90 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
108 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
22 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
37 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
63 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
243 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
144 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
327 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
61 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, ...