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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1answer
38 views

Dual basis in a finite separable extension

I am reading the book Algebras, Rings and Modules, volume 1, by M. Hazewinkel and at the page 193 there is a proof about why the integral closure of a ring in a separable finite extension L over $k$ ...
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1answer
18 views

Finding a basis and the dimension of $W_1\cap W_2$

Suppose $W_1,W_2$ are subspaces of $\mathbb{R}^4$. $W_1$ is spanned by $(1,2,3,4), (2,1,1,2)$ and $W_2$ is spanned by $(1,0,1,0),(3,0,1,0)$. I have to find a basis for $W_1\cap W_2$. I have ...
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1answer
67 views

Determining a basis of vector space with unusual addition and multiplications: $(a,b)+(c,d)=(ad+bc,bd) $ and $k * (a,b) =(kab^{k-1},b^k)$

Let $\mathbb W=\{(a,b) \in \mathbb R^2 \mid b>0\}$ and define addition by $(a,b)+(c,d)=(ad+bc,bd) $ and define scalar multiplication by $k * (a,b) =(kab^{k-1},b^k)$. Find a basis for $\mathbb ...
3
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1answer
297 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
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0answers
35 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 ...
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0answers
7 views

orbital behavior of objects in a space-time contraction field

I would like to find the orbital velocities of non-gravitating objects imbedded in a space-time contraction field. The field has the form as shown in figure 1. The surface space-time compression ...
2
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1answer
31 views

Modules isomorphism

Studying vector spaces, we can findthe well known result that every vector space of dimension $n$ over a field $k$ is isomorphic to $k^n$. Is there a similar theorem for modules? Thanks guys!
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2answers
42 views

Uniqueness of the solution to some differential equation.

I'm currently working on the subject mentioned in the title in a very general way. I think I get stuck for a stupid reason but here is my problem : I'd like to show that any solution to the equation ...
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1answer
36 views

A real vector space with a complex structure is naturally a complex vector space

I am struggling with this exercise from the book 'Tensors and Manifolds: With Application to Physics', by Robert H. Wasserman: Corresponding to each $a \in \mathbb{K}$ there is a linear operator ...
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1answer
39 views

Uniform continuity of scalar multiplication in topological vector spaces

If $X$ and $Y$ are topological vector spaces over $\mathbb R$, then a map $f:X\to Y$ is called uniformly continuous if for each neighborhood $V\subseteq Y$ of $0\in Y$, there exists a neighborhood ...
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5answers
529 views

Why do bases of infinite dimensional spaces need to be orthonormal?

I asked this question following a discussion in my Mathematical Methods course and didn't get a satisfactory answer. If we have an infinite dimensional Hilbert space, why do we need an orthonormal ...
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1answer
25 views

If M is a closed subspace of X and $x ∈ X-M$ then $M + \mathbb Cx$ is closed.

Let X be a normed vector space. If M is a closed subspace of X and $x ∈ X-M$ then $M + \mathbb Cx$ is closed. where $M + \mathbb Cx=\{y+\lambda x:y\in M, \lambda\in \mathbb C \}$ the question ...
3
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2answers
66 views

Equal-area sparse spherical shell partitioning

I'm trying to solve a particular problem that arose in a computer graphics context, but can be generalised to a bigger problem as well. I'm not entirely sure if this question belongs to MathExchange ...
1
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2answers
68 views

Vector spaces and dimension: unordered pairs

Let $K= \mathbb{Z}_p$, where $p$ is a prime number, and let V be a vector space over the field K such that $\dim{V} = 3$. I have no idea where to start with this, I'm not even really sure what I'm ...
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0answers
37 views

Arrow Space Construction

Is there a paper or book that has rigorously constructed the space of "arrow vectors" and shown that it is a vector space? By "arrow vectors" I mean oriented line segments in Euclidean n-space. ...
2
votes
0answers
36 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 $f_1(x) = 0, \ldots, f_p(x)=0$? ...
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1answer
48 views

Smallest angle between two vectors?

I have a robot and I am going to turn it clockwise (negative degrees) or counterclockwise (positive degrees). If I turn the robot -270 degrees, that is the same as turning +90 degrees. Is there a way ...
2
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2answers
27 views

Smallest angle to turn

I have a an object that starts an arbitrary heading in degrees. This object will rotate about an angle to reach a target heading. To reach this target heading, you can rotate about two different ...
3
votes
2answers
83 views

Kreyszig's Functional Analysis Section 2.8: How is the canonical embedding map injective?

Let $X$ be a vector space over the field $K$ of the real or complex numbers. Let $X^*$ denote the vector space of all linear functionals defined on $X$, and let $X^{**}$ denote the vector space of all ...
1
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1answer
34 views

Standard basis of a Matrix with identical entries.

How would you represent a $\mathbb{R}$-Matrixspace which looks like this $$\begin{bmatrix} a & b \\ b & a \end{bmatrix}$$ with standard basis I can't think of anything else but ...
1
vote
1answer
57 views

Show that $W$ is NOT a subsbace of $ \mathbb{R}^3$

$W $= {$(a,b,c):a^2 + b^2 + c^2 \leq 0$} As far as I have tried it with my concepts which are somewhat definitely not cleared the W is a subspace of $\mathbb{R}^3$ but I don't know why not. I tried ...
1
vote
1answer
260 views

Triangle and parametric coordinates

I'm studying on a book where it says: "A triangle is the set of points where for some point po, where u and v range over the parametric coordinates (we are talking about barycentric coordinates ...
0
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1answer
14 views

$A(2,1,3),B(3,2,4),C(1,8,9), D(4,3,12).$ Find the volume of a parallelepiped with vectors $\vec{AB}$, $\vec{AC}$, $\vec{AD}$.

$A(2,1,3)$ $B(3,2,4)$ $C(1,8,9)$ $ D(4,3,12)$ Find the volume of a parallelepiped with vectors $\vec{AB}$, $\vec{AC}$, $\vec{AD}$. I am not sure how to calculate this. How do I calculate the ...
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2answers
39 views

Vector space of dim n and its subspaces

Let $V$ be a vector space of dim $n$ over a finite field $F$ with $q$ elements. (a) Find the no.of dim 1 subspaces of $V$ (b) For each $1\leq k \leq n$, find the no.of dim $k$ subspaces of $V$ My ...
3
votes
2answers
52 views

If $K \leq L$ a finite extension then it is algebraic.

I am looking at the proof of If $K \leq L$ a finite extension then it is algebraic. The proof is the following: Let $[L:K]=n<\infty$. Let $a \in L$. We will show that $\exists$ a ...
2
votes
2answers
396 views

Exploring underdetermined linear system with non-negative solution

I haven't had much luck searching for this specific problems. Any pointers would be greatly appreciated. I have an underdetermined system where $ A $ and $ b $ are known. $ x $ is a real vector with ...
5
votes
0answers
67 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 ...
0
votes
1answer
34 views

Understanding normal and binormal of a vector or of a spline

I found a paper where it computes the 3D trajectory of a quadrotor and defines an error position as the difference between 2 vectors (here the source, under 3D trajectory control): $$ e_{p} = ...
6
votes
3answers
170 views

When is $V=U\oplus U^{\perp}$?

Let $V$ be a (infinite dimensional) vector space with inner product $(,)$ and $V$ may not be complete with the metric induced from the norm. Let $U$ be a subspace of $V$. What is the necessary and ...
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3answers
38 views

Understanding linearly independent vectors modulo $W$

We've learned in class: Let $W \subseteq V$, a subspace. $v_1, \ldots, v_k \in V$ are said to be linearly independent modulo $W$ if for all $\alpha_1, \ldots, \alpha_k: \sum_{k=1}^n \alpha_i v_i ...
9
votes
3answers
811 views

Linear independence of functions

I want to determine whether 3 functions are linearly independent: \begin{align*} x_1(t) = 3 \\ x_2(t) = 3\sin^2(t) \\ x_3(t) = 4\cos^2(t) \end{align*} Definition of Linear Independence: $c_1x_1 + ...
9
votes
3answers
253 views

An operation similiar to division of vector?

I need to solve these three equations to get $\vec x,\vec y,\vec z$ in terms of $\vec a,\vec b,\vec c$: $$\vec x \times (\vec y \times \vec z)=\vec a\\ \vec y \times (\vec z \times \vec x)=\vec ...
1
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1answer
59 views

Spaces of polynomials

Let $p_{1}(x)=-2x+2$, $p_{2}(x)=x+2$, $p_{3}(x)=x^{2}+2x+3$, $p_{4}(x)=x^{2}-x+3$. a) From the above four polynomials, determine a linearly independent subset that spans the polynomials. ...
2
votes
3answers
54 views

Whether the set of functions $(1,e^{x},e^{-x})$ linearly independent

Are the set of functions $(1,e^{x},e^{-x})$ linearly independent? I wrote it as an augmented matrix but it brought me to nowhere. Can somebody help me?
3
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4answers
70 views

How does $\dim \mathbb C$ work?

In the Wikipedia page about Dimension (vector space), it says the dimension of complex numbers is 2 or 1 if it's complex or real vector space respectively. How does that work? How to I describe ...
1
vote
2answers
45 views

Finding the basis and dimension of a vector space

Find the basis and dimension of vector space $ L_{1}$ spanned by vectors $ a_{1} ,a_{2},a_{3} $, the basis and dimension of vector space $ L_{2}$ spanned by vectors $ b_{1} ,b_{2},b_{3} $ and also ...
0
votes
0answers
28 views

If [x,z] = 0 $\implies$ [x,y] = 0, then y = $\alpha$z. True for infinite dimensional vector space?

I'm reading Halmos's Finite Dimensional Vector Spaces, in which he makes several references to the infinite dimensional case. In my edition this item appears as question 6 at the end of section 14. ...
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votes
2answers
70 views

$V \cong V \oplus V$ as $K$ vector spaces

I am not very sure about the triviality of this problem but I can't see the solution. Problem is If $V$ is a countable dimensional vector space over field $K$, then as $K$ vector spaces $V \cong V ...
1
vote
1answer
57 views

Symmetric algebra

If $V$ is a vector space over the field $K$ with basis ${v_1, v_2,…,v_n}$, then the symmetric algebra $S(V)= K[v_1,v_2,..,v_n]$. The question is: If $K$ is a commutative ring, then this equality is ...
0
votes
2answers
74 views

Co-planar unit vectors and their resultants.

Let $\bf a,b,c$ be three co-planar unit vectors such that $\bf a\hat \;b=\theta,b\hat\;c=2\theta$ where $\bf x\hat\;b$ is the angle between two vectors. Two vectors $\bf p,q$ are given by $$\bf ...
1
vote
1answer
49 views

Calculating a spread of $m$ vectors in an $n$-dimensional space

My question is regarding spreading $m$ vectors in an $n$ dimensional space such that the vectors are maximally distant from each other. For example, let us say I have a 2-D space, and 3 vectors, the ...
7
votes
2answers
339 views

Weird isomorphisms of infinite groups

According to my interpretation to one of the answers in Splitting in Short exact sequence, $$\Bbb R \cong \Bbb Q \oplus \Bbb R / \Bbb Q$$ also, according to What is known about the quotient group ...
0
votes
1answer
281 views

Orthogonal Complements and Subspaces Proof

I'm having a little difficulty understanding the proof for orthogonal complements. I kind of understand orthogonal complements, but I cannot seem to find a logic to this. I'm trying to follow along ...
0
votes
1answer
22 views

Inner product respect on a non-canonical base

Let a,b be vectors, on the standard base we use the dot product by simply doing a.b. But when we consider an other base we put a symmetric matrix between them. Why? How does that work? Thanks
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vote
1answer
39 views

The number of $(d-1)$-faces in a $d$-polytope is at least $(d+1)$

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset \mathbb{R}^d$ be a point configuration affinely spanning $\mathbb{R}$ (i.e., $\operatorname{aff}(V) = \mathbb{R}^d)$. Let $H$ be ...
2
votes
3answers
24 views

How much should I scale $dx$ and $dy$ individually to get a vector of required magnitude

I have a $dx$ and a $dy$ and I need to create a vector of magnitude $35.5$ in that $(dx, dy)$ direction. How much should I scale $dx$ and $dy$?
1
vote
1answer
46 views

Determining a basis for a space of polynomials

Determine a basis from the following set of second degree polynomials. Does this basis span the space of the second degree polynomials? What is the dimension of the (sub)space that it spans? ...
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votes
2answers
42 views

Whether a set of vectors span a subspace that includes a given vector

Do the vectors $(0, 1, 2), (1, 2, 1), ( -1, 2, 4)$ a) span $\mathbb R^{3}$ b) span a subspace that includes $w = (-2, 2, 10)$ I know they don't span $\mathbb R^3$ since they are ...
1
vote
3answers
35 views

Property of eigenvectors in linear mapping

Let $V$ be a bector space over a filed $\mathbb{F}$, and let $L:V\rightarrow V$ be a linear mapping. Let $U$ be a subspace of $V$ such that $L(U)\subset U$ Suppose that $u$ and $v$ are eigenvectors ...
3
votes
1answer
35 views

Prove dimension of sum of two subspaces

Let $U$ and $W$ be subspaces of $\mathbb{R^n}$ where $\dim(U)=n-1$, $\dim(W)=n-3$ and $n\geq 3$ Prove that $\dim(U\cap W)\geq n-3$ I used the property that both $U$ and $W$ are subspaces of ...