Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars

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1answer
24 views

Easy way to write standard basis for $F^n$?

Let $F$ be a field. The vector space $F^n=\{(a_1,\ldots,a_n):a_i\in F\}$ has a standard basis, where each of the vectors in the basis are $(1,0,\ldots,0)$, $(0,1,\ldots,0)$ and so on. Right now, I am ...
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0answers
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Generate distant n-dim points of numbers

I would like to randomly generate a set of n-dimensional points such that each point is far from all other points by a distance of $d$. In other words, I would like to have a set $S$ of n-dim points ...
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2answers
40 views

If $W \cong U \otimes V$, then what is $W$'s bilinear map?

Suppose that $U, V, W$ are vector spaces over a field $F$. Let $B : U \times V \to U \otimes V$ be the universal bilinear map (any bilinear map exiting $U \times V$ factors through $U \otimes V$ as $B$...
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3answers
31 views

Constructing a vector space of 8 elements from the field of 2 elements

I would like to use the field of 2 elements ($\mathbb{Z}/2\mathbb{Z}$) to construct a vector space consisting of 8 elements. I have succeeded in constructing vector spaces of 2 elements and 4 elements ...
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1answer
58 views

Dimension of $L(V,W)$ when $V, W$ are infinite dimensional?

For finite dimensional $V$ and $W$, we know that \begin{equation*} \dim{L(V,W)} = \dim{V}\cdot\dim{W} \end{equation*} Does this theorem hold for infinite-dimensional vector spaces $V$ and $W$ too?
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1answer
43 views

Correct way to say linear dependence/independence

$S=\{v_1,v_2,\ldots , v_n\}$ is a subset of a vector space $V$ over field $\mathbb{F}$. Which is the correct way to describe $S$? $S$ is linearly dependent/independent or vectors of $S$ are linearly ...
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1answer
31 views

Definition of basis for a subspace

Consider a vector space $V$ over a field $F$. According to my understanding, a basis of $V$ is a subset of $V$ that is linearly independent and spans the space $V$. Now consider a subspace $U\subseteq ...
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1answer
16 views

Fake rotation of an object in 3d space as if it was rotated by another point

If I have an object in a 3D world that rotates based off its center, how would I offset its position to fake or mimic as if it were rotated from another point? Example Image In this image, this cube ...
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2answers
30 views

Prove that operator norm of $X$ is smallest Lipschitz constant

Let$\left(U,\|\cdot\|_{U}\right)$ and $\left(V,\|\cdot\|_{V}\right)$ be normed Vectorspaces and $X: U \longrightarrow V$ linear and continous. $$ \|X\|:=\sup \left\{\|X(u)\|_{V}: u \in U \text { and }\...
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1answer
20 views

Relationship of transpose of linear mapping and Riesz isomorphism [closed]

How do I show the following: Be $Z$ a limited Euclidean vector space with the inner product space $\langle \cdot,\cdot \rangle$ and $\Phi : Z \to Z^*, z \mapsto \langle \cdot,z \rangle$ (Riesz ...
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1answer
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Basis, Alternative, Change of Basis

I'm trying to understand the practical purpose of using the basis vectors vs. alternative basis..and when you would want to/need to change the basis. Data Science related application would be ...
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0answers
45 views

The linear transformation satisfies $\sigma^3=\sigma^2-2\sigma$

Suppose $\sigma$ is a linear transformtation on linear space $V$, the linear transformation satisfies $\sigma^3=\sigma^2-2\sigma$ Prove that $V=\sigma^{-1}(0)\oplus \sigma(V)$ and if $\sigma^{-1}(0)\...
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20 views

Reference request for Counting subspaces and also for finite vector spaces

Can someone suggest me some good reference books and articles to study finite vector spaces I.e vector spaces over finite fields,and counting of number of subspaces,basis in such vector spaces.
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Let D([a, b]) be the collection of all differentiable functions on [a, b].

Let D([a, b]) be the collection of all differentiable functions on [a, b]. Show that D([a, b]) is a subspace of the vector space of all functions defined on [a, b].
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1answer
38 views

Dimension of subspace $W$ of vector space $V$ for infinite-dimensional vector spaces

Consider a subspace $W$ of a vector space $V$ (not necessarily finite-dimensional). I know that if $V$ was finite-dimensional, we can prove that $\dim W \le \dim V$. Does this hold even when $V$ is ...
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1answer
67 views

Any theory deals with spaces that mixes the size of the tuples?

Specifically, I am looking for for a theory that studies the properties of the set of a finite tuples of a set $X$. Suppose for instance the set of the reals, A 2-d space can by constructed as the ...
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30 views

Metric Linear Space

I'm not sure if this is the right place to ask and would appreciate it if someone directed me elsewhere if this is the wrong place to ask. I'm looking for exercises where I have to prove that a space ...
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0answers
25 views

Minimum number of inequalities characterizing a point

Suppose I have a point $x \in \mathbb{R}^d$. My question is, what the minimum number of inequalities $\phi_i^\top x \geq y_i$ is that is needed to uniquely characterize $x$, where $\phi_i \in \mathbb{...
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1answer
22 views

Given a linear transformation and basis, verify $ [T]_{\beta}^{\alpha}[v]_{\alpha} = T[(v)]_{\beta}$ . (More details in description)

Let $T : R^3 -> R^2 $ be the linear transformation defined by $T(x,y,z) = (3x +2y -4z, x-5y +3z)$, and let $\alpha$ = {(1,1,1), (1,1,0), (1,0,0)} and $\beta$ = {(1,3),(2,5)}. $Verify$ $ [T]_{\beta}...
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2answers
29 views

Question about dual spaces and subspaces

I don't understand this question at all.. this is the first time I've encountered anything regarding dual spaces and functionals and I'm completely lost. I know the dimension of $V^{*}$ is n, and that ...
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0answers
17 views

How to measure the similarity among vectors

Question Suppose there are three vectors $x$, $y$, and the target $t$. The angle between $t$ and $x$ is smaller than that of $t$ and $y$, but The distance between $t$ and $y$ is smaller that that of $...
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1answer
32 views

Checking for vector (function) subspaces

For the given Vectorspace V, one of the given sets is a subspace of V. Which one and why. Why does the other set not create a subspace? V = C(Real numbers), the Vectorspace of continouous Functions. ...
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0answers
24 views

Permutation of a Vector used to create a vector subspace (via Linear Span) of various dimensions

Let $v = (a, b, c, d)$ be a vector in $\mathbb{R}^{4}$. The components of $v$ can be arranged in exactly $24$ ways. For Example: $(a, d, b, c), (c, a, d, b), (d, c, b, a)$, and so on and so forth.... ...
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1answer
27 views

Characterization of continuity

So I found that this: $$\lim\limits_{h \to 0}\|L(x+h)-L(x)\|=0$$ implied continuity. Intuitively, I'd say it means that wherever you approach x from the limit is $L(x)$ but I struggle to see why. How ...
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1answer
34 views

Given 3 points. What's the normal to the plane that contains these 3 points?

Given are these points $$v_1=\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}, v_2=\begin{pmatrix} 2\\ 3\\ 3 \end{pmatrix}, v_3=\begin{pmatrix} 2\\ 4\\ 4 \end{pmatrix}$$ Determine the normal to the plane ...
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1answer
28 views

Show that $ f \circ g $ is self-adjoint iff $ f \circ g = g \circ f $ in a euclidean vector space.

This question has been asked before HERE, but I could not understand the following result from the linked post: $$ \langle F(G(v)), w \rangle = \langle G(v), F(w) \rangle = \langle v, G(F(w)) \rangle ...
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1answer
10 views

Compactness in trace norm.

Consider the space of trace-class operators $TC( \mathcal{H} )$ with the trace norm $\vert \vert \cdot \vert \vert_1$. Does it hold that the unit ball $ \{ \rho \in TC( \mathcal{H}) \mid \vert \...
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0answers
31 views

How do you prove that the determinant of a matrix is the same as the determinant of a transposed matrix? [closed]

Or differently phrased, how would you go about proving det(A) = det(A') This is as clear as it gets, I don't see why the question was closed. I know that they both have the same results, I am just ...
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1answer
26 views

Is $\mathbb{C^2}$ a vector space with new multiplication defintion over $\mathbb{C}$

I am trying to disprove this question: is $\mathbb{C^2}$ a vector space over $\mathbb{C}$ with regular addition and multiplication defined as: for every $\mathbb{z,w \in C^2}$ and every $\mathbb{\...
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0answers
25 views

Find [x]ẞ having two polynomial basis. [closed]

I have already found out the The change of basis from PA→B, however, I do not know, how to find [x]B without having a vector of basis A.
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0answers
20 views

nondegenerate symmetric bilinear form and direct sum

I need to determinate if this statement is true or false, some friends of mine think they found a proof but they are not sure meanwhile others seem to have a counterexample The statement is the ...
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0answers
12 views

Magma to construct a vector space

I'm aware this is an easy question. But I am new to Magma. So any help would be appreciated. let $K$ be the 27-dimensional complex vector space consisting of triples $m$ = ($m_1, m_2, m_3$) of complex ...
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1answer
44 views

Proving a vector space cannot exist

Hamilton tried to find a $3$-dimensional number system with the following properties: Every number can be written by $a + bx + cy$. This means every real number $a$ can be represented by $a + 0x + 0y$...
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0answers
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Euclidean distance as a sum of euclidean distances in projected subspaces

In [Ogras and Farhatosmanoglu, 2003] it is stated the following: Let us divide the $N$-dimensional space into $k$ orthogonal subspaces $S_1,\ldots,S_k$. Each subspace is of dimension $l=N/k$. Let us ...
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0answers
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Vector space in an sphere with a single isolated point [closed]

I am asked to find a vector space of a sphere (S^2(1)) such that it has just one isolated point, and give the index of the vector space in that point. I can't guess anyone. Thanks in advance.
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0answers
22 views

exceptional group via form preserving in magma

let $K$ be the 27-dimensional complex vector space consisting of triples $m$ = ($m_1, m_2, m_3$) of complex 3 x 3-matrices $m_i$, $1 < i < 3$, where addition and scalar multiplication are ...
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1answer
47 views

Classify all real $4 \times 4$ matrices satisfying $A^3 + A = A^2 + I$

This is an old prelim problem. The question is to classify all conjugacy classes of real $4 \times 4$ matrices satisfying $A^3 + A = A^2 + I$. Factoring this gives $(A-I)^2(A+I) = 0$. I can also see ...
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1answer
27 views

Ring homomorphism f from the field F into the endomorphism ring of the group of vectors - what for?

I am familiar with the definition of vector space. In the Wiki definition of vector space, https://en.wikipedia.org/wiki/Vector_space#Alternative_formulations_and_elementary_consequences. Wiki said: ...
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2answers
68 views

Why this endomorphism has n distinct eigenvalues?

Let $u$ be a diagonalisable endomorphism of $\mathbb{R}^n$. Let’s suppose that the family $(\operatorname{Id}, u, u^2,..., u^{n-1})$ is linearly independent and consider $\lambda_1, \lambda_2,..., \...
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1answer
23 views

Set of all sequences that have infinitely many coinciding elements? [closed]

Set of all sequences that have infinitely many coinciding elements are subspace of all sequences with real numbers? I am checking every axiom and it holds can you say what axiom does not hold here?
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All vectors that with given not zero $a$ vector do form a vector space? [closed]

For all set of vectors from origin that form with given non-zero $a$ vector $\alpha$ or $\pi-\alpha$ angle show they form subspace or not? ($0\leq \alpha \leq\pi$) I can't imagine how if we ...
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2answers
24 views

What are subspaces of $A$?

Can anyone please tell me what are the subspaces of $A$ ? My Thoughts: $\{\alpha I + \beta J : \alpha , \beta \in \mathbb R , \beta > 0 \}$ is not a vector space and it does not contain any non-...
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1answer
48 views

What common definition of norm on the space of analytic functions makes the basis $e_n=\frac{x^n}{n!}$ orthonormal?

I mean, this basis, with factorials is very useful as it is the basis of Taylor expansion. But it is not orthonormal under usual definitions of norm ($\int_a^b \sqrt{f(x)^2}dx$). I wonder, how it ...
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0answers
15 views

Inner (term-wise) product of two analytic functions, can it be expressed via integrals?

Suppose, $f(x)=\sum_{k=0}^\infty \frac{a_k x^k}{k!},$ $g(x)=\sum_{k=0}^\infty \frac{b_k x^k}{k!},$ which are standard Taylor expansions. What is the integral-based expression for $\sum_{k=0}^\infty \...
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0answers
25 views

A vector space basis with this strange property

I wonder whether anyone considered or proposed an infinite-dimensional vector space with such unusual property: an infinite sum over basis vectors with positive coeficients can produce a result, equal ...
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2answers
16 views

Calculated the distance between vector-line-equation and point. How do I find this point here?

I'm sorry for the weird title! I have a problem: Given is a point $p=\begin{pmatrix} 2\\ 2\\ 3 \end{pmatrix}$ and given is a line: $l(t)=\begin{pmatrix} 3\\ 3\\ 6 \end{pmatrix}+t \begin{pmatrix} 1\...
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1answer
38 views

Construct a matrix given the null space of A

Construct a matrix A with the null space spanned by the vectors $(2, -3, 1, 1, -1)^t, (1,0,-2,1,1)^t, (2,-2,1,0,-1)^t$ and $(-8,3,1,1,1)$ in $R^5$. I have come to the conclusion that the four vectors ...
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0answers
19 views

Subspace and its orthogonal in NSBF

Let $V$ a vector space of finite dimension on a field $K$, $W \subseteq V$ a subspace of $V$ and $f$ a scalar product on $V \times V$. Then $V=W \oplus W^⊥$. This proposition is true. I want to know ...
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2answers
43 views

Finding value of $m$ using vector dot product

Given the acute angle between the vectors $\textbf{a}=\binom{m}{0},\:\textbf{b}=\binom{1}{m}$ is thirty degrees, find the possible value of $m$ if $m$ is real. A simple dot product gives $$\frac{\...
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0answers
19 views

Finding an ortonormal basis for V={p∈P2|p(x) =a+bx2, a,b∈R} with innerspace 〈p,q〉=∫10p(x)q(x)dx. [closed]

Question says it all, i have tried finding similar problems online but cant find any

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