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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4
votes
5answers
119 views

Minkowski sum of two disks

An open disk with radius $r$ centered at $\mathbf{p}$ is $D(\mathbf{p}, r)=\{\mathbf{q} \mid d(\mathbf p, \mathbf q) < r\}$, and the Minkowski sum of two sets $A$ and $B$ is $A \oplus B=\{\mathbf p ...
2
votes
1answer
47 views

Computing dimension over a field of rationals

I am looking to find the dimension the vector space $V$ over $\Bbb Q$, the field of rationals, where the vectors are real numbers of the form $p + q\sqrt 2$, where $p$ and $q$ are rationals. I'm ...
1
vote
3answers
154 views

Prove an equality between dimensions of kernels

Let $V$ be a inner product space over field $\mathbb{R}$ with $\dim(V)<\infty$, and $T\in \text{Hom}(V,V)$. I'm trying to prove:$$\dim(\ker T)=\dim(\ker T^*)=\dim(\ker TT^*)$$ Also, as a conclusion ...
5
votes
2answers
144 views

Seeking an analytic proof of a vector identity

Show that for any vectors $\bf{u_1},\bf{u_2},\bf{v_1},\bf{v_2}\in\mathbb R^3$, we have $$(\bf{u_1}\times\bf{v_1})\cdot(\bf{u_2}\times\bf{v_2})= \left|\begin{matrix} \bf{u_1}\cdot\bf{u_2} & ...
0
votes
1answer
42 views

Show that $span({v_1,…v_n})=span({v_1,…,v_{n-1},w})$

This is what's given: $v_1,...v_n,w ∈ V$ and $v_1+...+v_n+w=0$ then I need to show that $span({v_1,...v_n})=span({v_1,...,v_{n-1},w})$ I could think of a way to show that this is true if I was sure ...
0
votes
1answer
161 views

What is $\mathrm{Sym}(V)$, where $V$ is a vector space

What is $\mathrm{Sym}^n(V)$, where $V$ is a vector space? (I've found a problem list and have some problems on notations)
0
votes
1answer
1k views

How to find rotation angles along X,Y,Z axes with a known vector to bring the axes to correct situation

I am working with 3d point data. When I checked the data I realized that there is some error on my data and need to do some kind of rotational rectification because the points which should be ...
1
vote
0answers
54 views

A simple piece of a Lemma on Gram-Schmidt

I was looking at a proof of Gram Schmidt theorem and I saw the following lemma, it starts here: First the theorem: if $V$ is an inner product space and $X= \{x_1,\dots, x_n\}$ is a linearly ...
2
votes
2answers
73 views

Is every linear representation of a group $G$ on $k[x_1,\dots,x_n]$ a dual representation?

Let $\rho\colon G\to GL(V)$ be a linear representation of $G$ on a $k$-vector space $V$. The dual representation is $$G\to GL(V^*),\quad g\mapsto(\varphi\mapsto\varphi\circ\rho(g^{-1})).$$ By the ...
0
votes
2answers
81 views

In an inner product space over $\mathbb R$, prove $ (u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \| $

Let $V$ be an inner product space over field $F$ and $u,w\in V$. Prove that if $F=\mathbb{R}$ then: $$ (u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \| $$ Is it also true for ...
-1
votes
1answer
66 views

Internally diving of vectors

Given the vectors $$\begin{eqnarray*}A&=&i+j-k\\B&=&i-j+2k\\C&=&j+k\end{eqnarray*}$$ How do I find the position vectors which divide BC AC internally in the ratio of 3:2?
2
votes
2answers
224 views

Is the unit disk in $\Bbb R^2$ a subspace?

This is the original Spanish version of the exercise: My understanding is that if we make a circle which has a center in [0,0] and a radius of 1 and we take all the points of that circle.... is ...
1
vote
1answer
72 views

Dual space of $K[X]$

Let $k[X]$ be the space of polynomials over a field $k$ (regarded as a vector space over $k$). What is the dual space of this vector space? My guess is that it is somehow generated by the derivations ...
1
vote
2answers
66 views

What is the mathematical term that can differentiate two same vectors?

Say i have two vectors A and B. Mathematically they are same if they have same magnitude and direction. So, say if someone asks me to draw a "vector" of 5 magnitude with 45 degree angle with ...
0
votes
4answers
71 views

A vector should more be thought an identity of an entity in space rathar than magnitude + direction?

Can I say that vector is more like a "unique identity" of an entity in space rather than calling it an entity with magnitude and direction ? For example a line. A vector $(10,10,0)$ is the identity ...
0
votes
0answers
151 views

how to get optimal vector, which is parallel to intersection line of many plane (Least Square way)

My idea is to construct the best optimal 3D line representing the intersection of many 3D planes. (As we know, due to fitting errors or data errors, the fitted planes might not intersect exactly ...
0
votes
1answer
33 views

Am I doing this Gram-Schmidt calculation correctly?

Here it is then, I want to check if I am doing this procedure the right way. We have V is a vector space $R_2[X] = ${$f|f=\lambda_0+\lambda_1X+\lambda_2X^2$} The inner product is defined as $VxV ...
0
votes
1answer
103 views

polar decomposition on finite dimensional vector spaces

Let $V$ be a finite dimensional inner product space on $\mathbb{F}$ (where $\mathbb{F}$ can be either $\mathbb{R}$ or $\mathbb{C}$) Let $A$ be a linear operator on $V$. The polar value decomposition ...
0
votes
1answer
2k views

Finding an orthonormal basis using Gram Schmidt process

OK, here's a question with polynomials. We want to find an orthonormal basis using Gram Schmift. Assuming that we are in a vector space V, $R^2[X]$ where {$f = \lambda_0+\lambda_1X+\lambda_2X^2$}. ...
0
votes
2answers
79 views

three dimensional cross product

Why do two three dimensional vectors $x$ and $y$ such that $x\cdot y$ does not equal $x\times y$ do not not exist? They do not exist right? Please help me kinda lost in this.
1
vote
1answer
522 views

Understanding a Gram-Schmidt example

Here's the thing: my textbook has an example of using the Gram Schmidt process with an integral. It is stated thus: Let $V = P(R)$ with the inner product $\langle f(x), g(x) \rangle = ...
0
votes
0answers
129 views

Forming the tensor product of a `real' vector space with a 'complex' vector space.

I have a question that I am hoping someone could clarify for me. Context: Consider the algebra $A = (B,\circ)$, given by: \begin{align} B = \{ \begin{pmatrix} a & f\\ \overline{f} & ...
1
vote
1answer
70 views

Is $U=\{(r,0,s)\mid r^2+s^2=0, r,s\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$?

Is $U=\{(r,0,s)\mid r^2+s^2=0, r,s\in \mathbb{R}\}$ a subspace of $\mathbb{R}^3$? If I set $r=s=0$, then it shows the zero vector is in $U$. For showing that U is closed under scalar ...
6
votes
2answers
221 views

why we want to use grassmannian space?

I wonder what's the special about grassmannian space? Why we want to use this space? On wikipedia, it says: "By giving a collection of subspaces of some vector space a topological structure, it is ...
0
votes
1answer
45 views

Largest angular distance from an orthonormal basis

Let $S^{n-1}$ be the unit sphere in $\mathbb{R}^{n}$. Let $\{e_1,\ldots,e_n\}$ be an orthonormal basis for $\mathbb{R}^{n}$. Let $\Sigma=\{e_1,-e_1,\ldots,e_n,-e_n\}$ be the set of $2n$ points ...
2
votes
2answers
55 views

Is $U=\{(r,s,t)|r,s,t \in \mathbb{R}, -r+3s+2t=0\}$ a subspace of $\mathbb{R}^3$?

Is $U=\{(r,s,t)|r,s,t \in \mathbb{R}, -r+3s+2t=0\}$ a subspace of $\mathbb{R}^3$? So far all I know is that the zero vector is in the subspace. How would I go about checking if it is closed under ...
2
votes
3answers
899 views

Dual space and inner/scalar product space

$V$ is vector space of finite dimension. $〈· , ·〉$ is an inner product on $V$.(Field $F$) We set transformation $T \colon V \rightarrow V^*$ as the following: $(T(v))(w) = 〈v , w〉$. Prove that $T$ ...
1
vote
2answers
364 views

$n=\dim V$. Then $V=\ker(T^n)\oplus\mathrm{range}(T^n)$

I trying to solve the following problem. The question is from a past exam. Suppose that $V$ is a finite dimensional vector space over a field $K$. Let $T: V\rightarrow V$ be a linear operator. If ...
0
votes
1answer
105 views

Vector spaces and finite dimensions related problem.

Please can you help me whit this problem. For $1.$ I did it as it's classical. What I am having trouble with are the other questions. Hints would be good but if you can explain that would be great. ...
1
vote
2answers
82 views

Linear algebra in Hilbert space

Let $M,N$ be closed subspaces of a separable Hilbert space. How to prove rigorously the following: $\operatorname{dim} M >\operatorname{dim} N => \exists u\neq0 \in M, u\in N^\perp$ ...
5
votes
3answers
225 views

What “is” a matrix in the context of a vector space?

I'm familiar with the definition of a vector space $V$ over a field $F$ I'm also comfortable with the notion that a matrix "represents" a linear map from one vector space $V$ to another vector space ...
0
votes
1answer
108 views

lagrangian subspace and Heisenberg group

Let $(V,\omega)$ be a symplectic vector space. Also we assume $L\subset V$ be a Lagrangian subspace., and $H(V)$ be Heisenberg group, then why $L\bigoplus U(1)\subset H(V)$ is maximal abelian ...
0
votes
2answers
54 views

Change of basis and identity

Let $\beta = \{b_1,\dots, b_n \}$ be a base for $V$. Explain why the $\beta$ coordinate vectors of $b_1,\dots, b_n$ are the columns $e_1, \dots, e_n$ of the $n$ by $n$ identity. The solution ...
1
vote
2answers
498 views

Proving a set of linear functionals is a basis for a dual space

I've seen some similar problems on the stackexchange and I want to be sure I am at least approaching this in a way that is sensible. The problem as stated: Let $V= \Bbb R^3$ and define $f_1, f_2, ...
1
vote
1answer
58 views

Questions on vectors

A line $L$ passes through the point with position vector $i-3j+2k$ and is parallel to the line $R=4i-5j+k+\lambda(5i+2k-k)$. a) Write down a vector equation for $L$. b) Hence find the ...
0
votes
2answers
67 views

Which of these sets is a subspace of F?

Let $F = \mathbb{R}^\mathbb{N}$. I need to check which of these sets are subspaces of $F$: $F_1 := \{ x \in F:\ \text{$x$ is bounded}\}$, $F_2 := \{ x \in F:\ \text{$x$ is convergent}\}$, $F_3 := \{ ...
2
votes
3answers
298 views

Prove that if $\langle x,z\rangle = 0$ for all $z$ then $x=0$

Hi I just wanted to check if my reasoning in this proof was correct. THe question is: let $\beta$ be a basis for a finite dimensional inner product space a) prove that if $<x,z> = 0 $ ...
4
votes
1answer
478 views

Proving that a Particular Set Is a Vector Space

Let $V$ be the set of all differentiable real-valued functions defined on $\mathbb R$. Show that $V$ is a vector space under addition and scalar multiplication, defined by $$(f+g)(t) = f(t) + ...
1
vote
1answer
140 views

Matrix of T, a linear transformation when Im T = Ker T

Let $V$ be a finite dimension vector space, $T:\ V \to V$ a linear transformation, and assume that $\operatorname{\rm Ker} T = \operatorname{\rm Im} T$. Prove that there is a basis $B$ of $V$, so that ...
2
votes
2answers
52 views

$K$ is a basis for $W$, and $L$ is a basis for $U$. Is $K\cup L$ is a basis for $U + W$?

Question: $V$ is a vector space over field $F$ , $U,W$ are subspaces of $V$. Is the next statement true or false? "$K$ is a basis for $W$, and $L$ is a basis for $U$. Therefore $K\cup L$ is a basis ...
2
votes
2answers
213 views

find symmetric line of given two line

I have one question. Suppose that we have two lines given by equations $$y=2x+3$$ $$y=-2x+11$$ I want to find all equations of lines which these two given lines have same distances from them ...
1
vote
1answer
127 views

Dependence of vectors : before and after linear transformation

I have a pretty simple question that confused me: V is a vector space of a finite dimension. $T: V \to V$ is a linear transformation. The information that's been given in question: ...
2
votes
1answer
225 views

Vectors Angles from $[0,2\pi]$

Given two vectors $V_1 = (x_1, y_1)$ and $V_2 = (x_2, y_2)$. How to calculate the angles between them in the range of $[0, 2\pi]$? I know the $\cos\theta$ similarity equation could present a $\theta$ ...
0
votes
1answer
68 views

Number of linear independent vectors multiples of other given their components have to sum to a fixed value.

My goal is to find, in any arbitrary dimension, how many linear independent vectors there are in the set made of all of the possible vectors in N whose components sum to a fixed value. $$A= ...
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 ...
1
vote
2answers
33 views

Only 'atomic' vectors as part of the base of a vector space?

Given a vector subspace $U_1=${$\left(\begin{array}{c} \lambda+µ \\ \lambda \\ µ \end{array}\right)\in R^3$: $\lambda,µ \in R$ } Determine a possible base of this ...
-1
votes
1answer
219 views

Positive linear combination of vectors to produce a positive vector

Given a list of vectors, I want a linear combination with positive coefficients that produces a final vector with only positive values (EDIT: this final vector is unknown; any positive vector is ...
0
votes
1answer
77 views

Get Normal of a 3D point.

I have set of points. I created strip triangles using these points. Now I need to calculate normal. What I thought that for each triangle there should be a normal. But function I am using says that ...
0
votes
2answers
150 views

How i can rotate a $m$ dimensional vector?

I have an $m \ge5$ dimensional vector $x$. How i can construct a rotation matrix $A$ s.t: $x \cdot A$ would be a vector rotated by $\theta$ angle. This is trivial for 2 dimensions, but what for this ...
2
votes
1answer
119 views

How to show $0$ is a point of closure of weak topology, but not a limit of weakly covergent sequence in a a subset of $l^2$

(von Neumann)For each natural number $n$, let $e_n$ denote the sequence in $\mathcal {l}^2$ whose $n$th component is $1$ and other components vanish. Define$$E = \{e_n + n \cdot e_m : n,m \in \Bbb ...