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
15 views

Calculating accelerometer vector when tilted

I'm trying to develop a 3D position estimation program using an xyz accelerometer. Ignoring the massive error introduced by double integration of the acceleration to get displacement, I have another ...
0
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1answer
10 views

Find vector resultant in rhombus

Uhm I can't find a solution for this problem, perhaps someone can help me with a hint or a solution, thanks in advance :) $$DG=GH=HI=IG\\and\\ AE=EF=FB$$ Find resultant for U+V+W
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0answers
36 views

Can someone please explain me in details the proof of the following theorem? Help

Let $(V,+,\cdot)$ be a vector space over a field $F$ and let $G$ be spanning for $V$ of size $n$. Let $L$ be a set of linearly independent vectors of $V$ containing $m$ vectors. Then $m\leq n$ and ...
0
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1answer
25 views

finite and infinite, vector space, linear transformation

I have to answer these questions for homework and I don't know if I'm answering these correctly. I think most of them are correct, but a double check would be much appreciated. a) If $S$ is a set ...
2
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1answer
33 views

Prove that if $v_1,v_2,…,v_r$ form a linearly independent set of vectors in $V$…

Let $S$ be a basis for an n-dimensional vector space $V$. Prove that if $v_1,v_2,...,v_r$ form a linearly independent set of vectors in $V$, then the coordinate vectors $(v_1)_S,(v_2)_S,...,(v_r)_S ...
2
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3answers
35 views

Prove that if $\dim X'<\infty$ then $\dim X<\infty$

I have to prove that $\dim X'<\infty$ then $\dim X<\infty$ where $X$ is a normed vector space and $X'$ is a space of all linear and continuous functionals from $X$. How can I prove this? I ...
0
votes
3answers
44 views

Show that if the set $\{v_1,v_2,v_3\}$ is linearly independent then so are all subsets. [on hold]

Show that if the set $\{v_1, v_2, v_3\}$ is linearly independent then so are $\{v_1, v_2\}$, $\{v_1,v_3\}$, $\{v_2,v_3\}$, $\{v_1\}$, $\{v_2\}$ and $\{v_3\}$. I don't even know what to start with. Do ...
3
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1answer
23 views

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$.

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$. Let consider $H=l_2$ where $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace $ ...
0
votes
2answers
30 views

Proof that if we add a vector to a linearly dependent set of vectors in a vector space $V$, then the new set of vectors is still linearly dependent

Prove that if $S=\{v_1, v_2, v_3\}$ is a linearly dependent set of vectors in a vector space $V$, and $v_4$ is any vector in $V$ that is not in $S$, then $\{v_1, v_2, v_3, v_4\}$ is also linearly ...
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2answers
16 views

Orthonormal basis of $\mathbb{C}^{r}$

I have to prove that $\{f_1,f_2,..,f_r\}$ is an orthonormal basis of $\mathbb{C}^{r}$ where: $f_j=\frac{1}{\sqrt{r}}(1,e^{2i\pi\frac{j-1}{r}},e^{4i\pi\frac{j-1}{r}},...,e^{2(r-1)i\pi\frac{j-1}{r}})$ ...
0
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0answers
7 views

Subspace invariant under irreducible Coxeter group

I'm trying to show that if $G$ is an irreducible Coxeter group, then it acts irreducibly on vector space $V$. That is, $V$ has no nontrivial $G$-invariant subspaces. I started by assuming that $V$ has ...
2
votes
3answers
64 views

Find dimension of ℒ $(V)$ and polynomial that brings every linear transformation to $0$

Here's the prompt: Let V be a vector space of finite dimensions $n$ over the field $\mathbb{F}$, and let $\tau \in$ ℒ $(V)$. What is the dimension of ℒ $(V)$ as a vector space over $\mathbb{F}$? With ...
1
vote
1answer
18 views

Find a vector $t \in \{x,y,z\}$ with base $\{u, v, w\}$

I don't know how to find a vector $\vec t$ that will suffice the condition: $\vec t \in \{x,y,z\}$ with bases $\{u, v, w\}$ the given vectors are: $$ \begin{array}{rcrrrrrl} u &=& [ & ...
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0answers
13 views

Show that a plane in R^3 is a vector space over R if and only if the origin lies in the plane.

Show that a plane in $\mathbb{R}^3$ together with the usual addition and multiplication is a vector space over $\mathbb{R}$ if and only if the origin lies in the plane. (Hint: you may use that ...
0
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1answer
37 views

Why this set is not a vector space?

Let V =R^2 and define addition and scalar multiplication operation as follows : $u=(u_1, u_2)$ $v=(v_1, v_2)$ $$u+v=(u_1+v_1,u_2+v_2)$$ $$ku=(u_1k,0)$$ The book says : "the addition operation is ...
0
votes
1answer
25 views

Why $\dim(\hom(V,W))=\dim(V) * \dim(W)$?

I have found that the question I want to ask someone had asked, here is the website: $\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$ Here is my question: Why $\dim(\hom(V,W))=\dim V * \dim W $? ...
2
votes
1answer
28 views

What is meant by $\langle \cdot,\cdot \rangle ^H_\mathbb{R}$?

there is the following statement: Let $\langle \cdot,\cdot \rangle_\mathbb{R} = \sum_{k = 1}^{n} x_k y_k$ be that standard Euclidian scalar product in $\mathbb{R}^n$ and $\langle \cdot,\cdot ...
2
votes
2answers
63 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
0
votes
1answer
31 views

Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
1
vote
1answer
45 views

Prove that the mapping $\psi : L(V,W) \rightarrow L(W^*, V^*)$ given by $\psi(T) = T^t$ is an isomorphism.

Let $V,W$ be finite-dimensional vector spaces over the same field $\mathbb{F}$ and let $L(V,W)$ be the vector space of $\mathbb{F}$-linear transformations from $V$ to $W$. Prove that the mapping ...
1
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2answers
16 views

concept between group and vector space, compare G/N with V/W

When we considered factor groups G/N, we need N to be normal,but in vector space V/W, why W only be subspace?
2
votes
1answer
29 views

Characterization of separability in normed vector spaces.

I want to prove the following statement: In a normed vector space $(V,\|\cdot\|)$ it holds that $$ V \ \text{separable} \ \Longleftrightarrow \ \{ v \in V \colon \|v\|=1\} \ \text{separable}.$$ I ...
1
vote
1answer
16 views

Calculate the unit normal vectors to the both sides of a plane

Calculate the unit normal vectors to the both sides of a plane passing through three points with coordinates (1,0,1), (1,1,-1) and(-1,1,1). My answer is [$\sqrt{6}/6 , \sqrt{6}/3 , \sqrt{6}/6$] and ...
1
vote
2answers
19 views

Determine if all vectors of the form (a,b,c), where b=a+c+1 are subspaces of R^3?

Determine if all vectors of the form $(a,b,c)$, where $b=a+c+1$ are subspaces of $\mathbb{R}^3$? Use the theorem: If $W$ is a set of one or more vectors from a vector space $V$, then $W$ is a ...
0
votes
1answer
60 views

inner product space , dual space, proof about isomorphism

Let $V$ be a vector space (not necessary being finite dimensional) and let $U,W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $V^\ast/(W^0)$ is isomorphic to $W^\ast$. Notation and ...
0
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0answers
10 views

Get Attitude from 2-axis vector

I've built a quadrotor but my 3-axis accelerometer has a fault, the Z-Axis doesn't work. I would normally get my attitude with the following code pitch = atan2(accel_X, accel_Z)*RadToDeg; roll= ...
1
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1answer
13 views

equality of annihilators of a vector space

I have a problem as follows: W1 and W2 are subspaces of a finite-dimensional vector space V. W0 is the annihilator of W. (a) Prove W01=W02 implies W1=W2
0
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1answer
33 views

V = U⊕W then Prove that (V/W)* is isomorphic to W^0

Let $V$ be a vector space (not necessary being finite dimensional) and let $U$, $W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $(V/W)^*$ is isomorphic to $W^0$. note: (V/W)* is the ...
3
votes
1answer
23 views

Why should we expect the divergence operator to be invariant under transformations?

A lot of the time with vector calculus identities, something that seems magical at first ends up having a nice and unique proof. For the divergence operator, one can prove that it's invariant under a ...
0
votes
1answer
22 views

Prove Linear Dependence in T: V -> W

Problem: "Let $V$ and $W$ be vector spaces and let $T:V \rightarrow W$ be a linear transformation. Prove that, if $\{v_1, v_2, v_3\}$ is a set of three linearly dependent vectors in $V$, then the set ...
0
votes
0answers
25 views

Why is $\hat{x}$ in the linear regression equation $A^TA\hat{x} = A^Tb$ part of $C(A^T)$

When finding the best fit line for a number of points, we use $A^TA\hat{x} = A^Tb$ where we solve for $\hat{x}$. I understand that the projection $p=A\hat{x}$ is part of the column-space of $A$ and ...
2
votes
1answer
19 views

How to get perpendicular vector close to another vector?

Suppose I have two vectors $\vec v_1$ and $\vec v_2$ in $E^3$ space. How can I find a vector $v_3$ such that $\vec v_3$ is perpendicular to $\vec v_1$ the angle between $v_2$ and $v_3$ is minimized ...
1
vote
3answers
69 views

An infinite generating set of a finite dimensional vector space contains a basis

Let $S$ be an infinite generating set of a finite dimensional vector space , then how do we prove that there is a subset of $S$ which is a basis of the vector space ? Please help
1
vote
2answers
18 views

Finding an Orthonormal Basis using Gram Schmidt

Given the set of vectors $S=${${V_1=\binom{1}{4},V_2=\binom{4}{-4} }$} I am to find an orthonormal basis for $R^2$ using the Gram-Schmidt process. I've already worked it out and found the orthonormal ...
0
votes
0answers
8 views

How to find optimal perpendicular axis of rotation vector?

I am drawing lines on the screen. Each line has a point (x,y,z) and a direction (u,v,w). I want to draw arrow heads on these ...
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votes
1answer
41 views

Exercise 3.6: Elementary Functional Analysis By Barbara [closed]

Let $X=\ell_\Bbb R^\infty$ denote the space of bounded sequences with real entries, in the supremumnorm. Consider the operator $T$ defined on $X$ by $T(x_1, x_2, . . .) = (x_2, x_3, . . .)$; this is ...
2
votes
1answer
19 views

Defining operations for a vector space

I was hoping someone could help me with the following. Is it possible to define operations + and $\cdot$ on this set to make it a vector space: \begin{equation*} ...
0
votes
1answer
19 views

The difference between norm and modulus

I'd like to know the difference between norm of a vector, ||v|| and the modulus of a vector, |v|
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0answers
3 views

Gentle introduction to discrete vector field [closed]

I am looking for a gentle introduction to discrete vector field. Thanks in advance.
0
votes
1answer
45 views

Question on sequence space (as a linear space)

Let $X$ be the space $\ell_\infty$ of all bounded sequences of real scalars. If $Y$ is the set of all $x\in X$ that have bounded partial sums (1) Can I say $Y$ is a linear space (as a subspace of ...
3
votes
2answers
41 views

Arbitrary (i.e. not necessarily finite-dimensional) vector spaces; reference request.

Its virtually impossible to complete an undergraduate degree these days without studying finite-dimensional vector spaces in quite some detail. So like most of us, I've done all that; however, just ...
1
vote
3answers
29 views

A question on basis of vectorspaces and subspaces

Let $V$ be a finite dimensional vector space and $W$ be any subspace . It is known that if $A$ is any basis of $W$ then by "extension-theorem" , there is a basis $A'$ of $V$ such that $A \subseteq ...
-1
votes
1answer
53 views

How to draw arrows by rotating lines in 3d space?

I am trying to figure out direction vectors of the arrowheads of an arrow. Basically I'm given a normalized direction vector ...
1
vote
1answer
34 views

Determining and enforcing linear dependence

Assuming we have a large set of multi-dimensional vectors (20k vectors, 100 dimensions each). My questions are the following: How can we determine the level of linear dependence of this set? Is ...
0
votes
0answers
11 views

Transformation Matrix of a function

I have the following: (Note: $V^{*}$ is defined as: $V^{*} = \{ L: V \rightarrow \mathbb{R} | \text{L is linear} \}$) Let $V$ be an $\mathbb{R}$-Vectorspace. Let $\phi \in V^{*} \text{ \ } \{0 \}$ ...
0
votes
1answer
26 views

Vectors and Planes

Let there be 2 planes: $x-y+z=2, 2x-y-z=1$ Find the equation of the line of the intersection of the two planes, as well as that of another plane which goes through that line. Attempt to solve: the ...
0
votes
0answers
5 views

Geometry of Vectors question

Let there be 3 points; $A(1,2,0), B(2,2,-1), C(4,0,1)$. Find the plane in which all $3$ lie, and find point $D$ such that $ABCD$ is a parallelogram . I did find the plane equation which is ...
1
vote
1answer
32 views

What is the canonical basis of a dualspace in $\mathbb{R}^3$?

I have the following: Consider the basis $$B := \{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \}$$ of the ...
0
votes
0answers
25 views

Vector Question Help

A plane is determined by $(x,y,z) = (1,-1,0) + t(1,-1,2)$ and point $p(1,2,3)$. find point of intersection of $(1,4,-1)+s(-6,2,-4)$ with this plane. I tried this: given the data plane equation is:$$ ...
1
vote
1answer
26 views

Why we take transpose of Vector (Displacement Vector)?

I'm trying to understand some equations that involves transpose of vectors (displacement vectors to be precise) Two set of vectors F and G (with i,j) that corresponds to X,Y value in plane and ...