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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2
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Proving a subset is a subspace of a Vector Space

To prove a subset is a subspace of a vector space we have to prove that the same operations (closed under vector addition and closed under scalar multiplication) on the Vector space apply to the ...
0
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3answers
96 views

Proof needed for this exercise from “Linear Algebra Done Right”

Suppose that $U$ and $V$ are finite-dimensional vector spaces and that $S\in \mathcal{L}(V,W)$ and $T\in \mathcal{L}(U,V)$, where $\mathcal{L}(X,Y)$ is the vector space of linear transformations from ...
2
votes
1answer
29 views

Partial derivative or something else?

In the formula for the Reimann tensor Wikipedia says that $$∂_μ=\frac{∂}{∂x^μ}$$ and that they are coordianates of a vector field. But does it just mean the partial derivative of what comes after is ...
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0answers
45 views

Relation between componenet and algebraic definition of covariant vectors

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ...
0
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2answers
24 views

Position of a point with respect to two reference frames

I working on a project where doing some image processing detect objects using Kinect camera and then move it to a desired location with a help of robotic arm. In this project the sensor gives pixel ...
3
votes
3answers
47 views

Not a basis for $ l^\infty$ then what is it?

We know that $ l^\infty$ has not a Schauder basis and its Hamel basis is uncountably infinite. Let $e_n=(e_{n1}, e_{n2},...)$ (for each $n\in \mathbb{N}$) s.t. $e_{nj}=0$ when $n\neq j$ and ...
4
votes
2answers
62 views

An example of space $V$ such that $(V^{\perp})^{\perp} \neq V$

I know that if $W$ is a vector space of finite dimension then for any subspace $V$ ,$(V^{\perp})^{\perp} = V$. But I have heard that this is not true for infinite dimensional vector spaces. So I tried ...
1
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1answer
19 views

parametric form of vector

I am having trouble understanding what the question is asking at this point, I have solved the first parts correctly and was wondering if I could get help as to how to solve x=x(t)
0
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3answers
36 views

magnitude of two vectors

How would I find the crossproduct if all I have is the point values?
0
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2answers
21 views

plane which passes through three points

I am confused as to how to answer this question because I don't understand how to incorporate the 12 into my answer. Any suggestions?
0
votes
1answer
34 views

Help with notation in linear algebra

Here is a question from a final I am doing I need help with understanding what exactly it means with $p(0)$ or $p(1)=0$ mean. I know that $p(x)=c_0+c_1x^1+c_2x^2+...+c_nx^n$ so does that mean ...
1
vote
3answers
56 views

What is the base for the subspace defined by: $F= \{(x,y,z) \in \mathbb{R}^3 | x − y + z = 0\}$? [closed]

What is a basis for the subspace defined by $$F= \{(x,y,z) \in \mathbb{R}^3 \mid x − y + z = 0\}?$$
2
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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 ...
0
votes
0answers
14 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
1
vote
1answer
43 views

Dimension of a vector space.

Let $v_1,v_2,v_3,v_4$ and $v_5$ be the non-zero vectors of a vector space $V$ such that $a_1v_1+a_2v_2+a_3v_3+a_4v_4+a_5v_5\neq0 \hspace{1cm} (\forall a_i\neq0,\, 1\leq i\leq5)$ Then what is the ...
6
votes
2answers
150 views

Dimension of R over Q without cardinality argument. [duplicate]

I am looking for the easiest (elementary) proof that $\mathbb R$ is infinite dimensional as a $\mathbb Q$-vector space, without using cardinality. It should be understandable at highschool level. So ...
0
votes
0answers
65 views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis i , j , k so by invariance nature of vectors, component of gradient ...
0
votes
0answers
27 views

linear algebra question related to basis, kernel and linear transformation [duplicate]

Let V be a 2-dimensional vector space, and let α=e1,e2 be a basis for V. Define a linear transformation T:V→V by declaring that: T(e1+e2)=2e1−e2 T(e2)=4e1−2e2. a. Find [T]α,α. (one alpha is upper ...
1
vote
1answer
45 views

having trouble with a change of basis of a linear transformation.

Let $\Bbb V$ be a $3D$ vector space with a chosen basis $\alpha=\{e_1,e_2,e_3\},\ \beta=\{f_1,f_2,f_3\}$ for $\Bbb V$ satisfying: $e_1=f_1+f_2+f_3$ $e_2=f_2+2f_3$ $e_3=f_3$ find the four ...
2
votes
1answer
31 views

Kernel of a map into infinite dimensions

I'm trying to calculate the kernel of a linear map, but the codomain is infinite dimensional and I'm not sure if there's something that I'm missing. Let $V$ be a (two-dimensional, with basis ...
0
votes
1answer
70 views

Having trouble with a linear lager question about kernel and basis

Let $V$ be a $2$-dimensional vector space, and let $\alpha={e_1,e_2} $ be a basis for $V$. Define a linear transformation $T: V\to V$ by declaring that: $T(e_1+e_2)=2e_1−e_2 $ $T(e_2)=4e_1−2e_2$. ...
0
votes
1answer
30 views

Coordinate Matrix

I am really struggling with the concept of coordinate vectors and hence coordinate matrix in vector spaces. It would be great if anyone could provide me any intuitive picture to understand it.
-3
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2answers
41 views

What is meant by an orthonormal basis and orthogonal basis for a vector space? [closed]

And how are these two things different from each other.
1
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0answers
38 views

having trouble with a 3-dimensional basis-change problem/

Let $V$ be a 3d vector space with a chosen basis $\alpha=\{e_1,e_2,e_3\}, \beta=\{f_1,f_2,f_3\}$ for $V$ satisfying: $$\begin{align}e_1 & =f_1+f_2+f_3 \\ e_2 &=f_2+2f_3 \\ e_3 & =f_3 ...
0
votes
2answers
38 views

i am having trouble with one of the homework question regarding to linear algebra(vector and span)

$V$ is a vector space of some dimension, with $\vec u,\vec v,\vec w$ independent set of vectors in $V$. define the subspace of $V$ given by $W = \operatorname{span}(\vec u-\vec v+\vec w, 2\vec u+\vec ...
2
votes
2answers
45 views

closeness of a set of vectors

Is there some measure that captures the "closeness" of a set of vectors? Say I have a matrix, $$ A = \left[ \begin{matrix} 0.8 & 0.15 & 0.05 \\ 0.82 & 0.09 & 0.09 \\ 0.78 & 0.08 ...
1
vote
2answers
34 views

Properties of the vector space $V^-$ defined by “refusing” to multiply the complex vector space $V$ by anything other than real scalars.

I$\def\nc#1#2{\newcommand{#1}{#2}}\nc{\vm}{V^-\!}\nc{\v}{V}\nc{\f}{\mathbb{F}}\nc{\fn}{\f^n\!}$ am reading "Finite Dimensional Vector Spaces" by Halmos and several times he mentions the space denoted ...
0
votes
2answers
64 views

How do you find a non zero vector in Linear Algebra?

The question is; The vectors $a_1 = (1, 1, 0)$ and $a_2 = (1, 1, 1)$ span a plane in $\Bbb R^3$. Find the projection matrix P onto the plane, and find a nonzero vector $b$ that is projected to zero. ...
0
votes
0answers
25 views

Finding the least square fit for 3 parameters in Linear Algebra

I know how to find least square for $y = mx+b$ when we have two parameters. But this question has $3$ parameters, am trying to think of how to approach it but so far no success, I can't find any ...
1
vote
1answer
28 views

Direct Sum of Three Subspaces

Suppose $U = \{(x, y, x+y, x -y, 2x) \in \Bbb F^5 : x, y \in \Bbb F\}$. Find three subspaces $W_1, W_2, W_3$ of $\Bbb F^5$, none of which equal $\{0\}$ such that $\Bbb F^5 = U \oplus W_1 \oplus W_2 ...
0
votes
2answers
49 views

How can vectors with different units (position, speed, …) “share” the same space?

My question is too stupid to be googled, so I'll ask it here (because I didn't get any answers from google). Context: I have a three dimensional space and the units for $x,y,z$ are given in meters. ...
0
votes
0answers
20 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
1
vote
1answer
10 views

Finding the conditions of (x,y,z,t) for them to belong to the span of a set of vectors

So I got this math exercise, and I don't know how to go about it: In $\mathbb{R}^4$, $S$ is the subspace spanned by the following set of vectors: $(1, 1, 1, 0) , (1, 2, 1, 1) , (2, 0, 1, 1) , (3, 0, ...
0
votes
1answer
40 views

Coordinate matrices in standard basis

If $A=1+2x+4x^3$ and $B=2+3x^2+x^3$ are vectors in Polynomial space, find out the coordinate matrices for $A$ and $B$ in standard basis and hence find out the angle between vectors A and B.
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0answers
25 views

Is there a computationally efficient way to find the part of a vector, which is of certain order in independent variable x?

Let $\vec{a}$ be an element of a vector space over the space of monomials, i.e. $$ \vec{a}\left(x\right)=\sum_{j=1}^{N}a_jx^{k_{j}}\vec{e_{j}} $$ Remark: For simplicity, here we operate with only ...
0
votes
3answers
52 views

How to prove a levi-civita symbol and kronecker delta relationship [duplicate]

$\displaystyle \sum_{i=1}^3 \sum_{j=1}^3 \epsilon_{ijk} \epsilon_{ijn} = 2 \delta_{kn}$ When I do the calculations of that I get 3 times the answer, I mean this is easy, but I´m just wrong, Could ...
0
votes
0answers
28 views

$\dim \mathcal{S}_k(\Gamma_0(N))$

I'm looking for a formula which gives the dimension of $\mathcal{S}_k(\Gamma_0(N))$ the space of cusp forms of weight $k$ and level $N$. I found the following statement for $k\geq 4$ $$\dim ...
0
votes
1answer
32 views

Finding the exponential relation between two 4x4 transition matrices

Im alright with matrices, but this question has dumb-struck me. Suppose I have two known and given $4\times4$ transition matrices, representing transitions in three dimensions with the fourth ...
1
vote
2answers
22 views

Prove that addition of a constant on vector spaces is bijective

What would be a nice way to deduce from the vector space axioms that $f : V_1 \longrightarrow V_2, \, x\mapsto x+v$ with constant $v$ is bijective?
0
votes
0answers
14 views

Projection of vector onto intersection of two planes

Given that $U$ is a vector space in $\mathbb{R}^{n}$ there exists a unique vector $\mathbf{u_{0}}$ such that $$\left \| \mathbf{v}-\mathbf{u_{0}} \right \|\leq \left \| \mathbf{v}-\mathbf{u} \right ...
0
votes
0answers
14 views

Properties of cross product ${\rm i}(a\times a^*)$

Given a complex 3-vector $a\in\mathbb{C}^3$, let $b$ be the following vector $$b={\rm i}(a\times a^*)$$ where $a^*$ is the element-wise complex conjugate of $a$. As can be easily shown by ...
2
votes
1answer
17 views

Vector space and its Projecctivized Space

Why is the co-dimension one subspaces are the points of $\mathbb P(V^{\vee})$. $V^{\vee}$ is the dual space of V and and $\mathbb P(V)$ is the projectivized space of V. $\mathbb P(V)= ...
0
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0answers
24 views

Do the Generalized Gell-Mann Matrices form a complete set?

Please bear with me, I'm studying Lie algebras as they are related to quantum mechanics, and most of my group theory knowledge is self-taught. I'm not sure how to prove this seemingly basic result. ...
2
votes
2answers
43 views

Dimension of vector space in extreme cases.

Let V be a vector space of dimension 29 over a field $\mathcal{F}$. Suppose that U and W are subspaces of V with dim(U) = 24 and dim(W) = 15 1) What are the possible values of dim(U+W)? My ...
5
votes
2answers
96 views

Localization does not commute canonically with infinite direct products

Let $S=\mathbb{Z}-\{0\}$, and the fraction ring \begin{equation} S^{-1}\prod_{1}^{\infty}\mathbb{Z}_{i}=\{\frac{(a_{1},a_{2},...,a_{n},...)}{b}:b,a_{i}\in\mathbb{Z},b\neq 0\}.\end{equation} Show ...
2
votes
0answers
33 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 ...
1
vote
0answers
26 views

Getting coordinate vector in linear algebra

I know how to get the coordinate vector of single matrices by just joining them and doing a gauss jordan. But these are a 2x2, I don't know how to go about this, apparently no elimination can take ...
3
votes
1answer
81 views

How many (unordered) bases does $\Bbb F_q^n$ have as a vector space over $\Bbb F_q$?

Following the recommendation here to get this question out of the unanswered queue, I've changed this from a proof-verification question into an answer-your-own. Here's the question again in case ...
0
votes
2answers
30 views

vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
0
votes
1answer
20 views

How to show a vector space is not closed under addition with elements not in the vector space.

Wasn't entirely sure how to word the title. What I'm trying to show is: Given $\vec{v}\in V$ and $\vec{w}\not\in V$, then $\vec{v}+\vec{w}\not\in V$ How would this statement be proven?