Questions tagged [hamel-basis]

A Hamel basis (often simply called basis) of a vector space $V$ over a field $F$ is a linearly independent subset of $V$ that spans it.

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Deriving a basis from a generating subset and a free subset

Let $K$ be a field and $E$ a $K$-vector space. Let $S$ be a generating system $E$ and $L$ a free subset of $E$ such that $L\subset S$. Then there exists a basis of $E$ such that $L\subset B\subset S$. ...
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Show linear independence of sum of vectors from a base

let $V$ be a vectorspace and $\{a_1,a_2,a_3,a_4\}$ a base of $V$; Moreover $b_1:=2a_1-a_2,$ $b_2:=a_2+a_3+a_4,$ $b_3:=a_3-a_4$; Show that $b_1,b_2,b_3$ are linear independent; By definition of linear ...
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Is a basis for the vector space of all series in $\mathbb{R}$ constructible [duplicate]

Given the $\mathbb{R}$-vectorspace $V =\mathbb{R}^\mathbb{N}$ of real valued series I was wondering if we can construct a Hamel-basis of $V$. First of all I think that the dimension of $V$ is $ \mid\...
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Basis for complement of a subspace

Consider we have a vector space $V$ with a given basis $\{v_i\}_{i \in I}$, and a subspace $U$. Show that we can always find a subset $J \subset I$ such that $\{v_i\}_{i \in J}$ is a basis for a ...
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Cardinality of Hamel basis of a separable incomplete inner product space

I know that cardinality of Hamel basis of an infinite dimensional separable Hilbert space is always equal to the cardinality of the continuum. But if an infinite dimensional separable inner product ...
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Show that the set of the given vectors form a basis in $\mathbb R^3$ and represent the standard basis as a linear combination of these vectors

Show that the vectors $\langle 1,0,-1\rangle$,$\langle 1,2,1\rangle$,$\langle 0,-3,2\rangle$ from a basis in $\mathbb R^3$. Represent standard basis in $\mathbb R^3$ as a linear combination of the ...
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Kreyszig's definition of finite dimensional vector spae

In his book on functional analysis Kreyszig gives the following definition: Definition. A vector space $X$ is said to be finite dimensional if there is a positive integer $n$ such that $X$ contains a ...
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What is the dimension of a vector space without a basis?

It's my understanding that without the axiom of choice, the vector space of $\mathbb{R}$ over $\mathbb{Q}$ lacks a set of basis vectors. Now it seems straight forward that with the axiom of choice ...
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Vector space T-invariant

I'm studying the problem Prove that $W$ is $T$-invariant if and only if $W^0$ is $T^t$-invariant., but I'd like to solve it considering V infinite-dimensional. I know that all vector space has a Hamel ...
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Show that absorbing and convex set contains Hamel basis

Show that absorbing and convex set contains Hamel basis. I know the definitons of convexity and being absorbing. Also, I know that every vector space has Hamel basis, but these are all informations I ...
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prove that there are infinitely many additive functions on $\mathbb{R}$ which are not linear

I was reading chapter 4 of the book Functional Equations by B.J. Venkatachala (Page No. $118$) and found a remarkable proof to the fact that there are infinitely many additive functions ($f:R \to R$ ...
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Example illustrating the importance of Hamel bases for description of additive functions

Can anyone please explain and show (step by step) that how $ f \big( f ( x ) \big) = x $ in the following example? One can show that for any Hamel basis $ H $ there exists a bijection $ \varphi : [ 0 ...
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Importance of Hamel bases for description of additive functions

I am unable to understand the reason and motivation of the author that why did he set $$ f ( x ) = \sum r _ \alpha s ( h _ \alpha ) \text . $$ The importance of Hamel bases for description of ...
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What is the dimension of the vector space consisting of all real-valued functions?

The dimension of this vector space is obviously infinite dimensional, and it's not too much work to show that its basis is an uncountable set, making it an uncountably-infinite dimensional vector ...
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Why from the point of view of constructive analysis denumerable hamel basis is not suitable for an infinite dimensional Banach Space

We know that for an infinite dimensional Banach Space the Hamel Basis needs to be uncountable. Why is it difficult from the point of view of constructive analysis to obtain an Uncountable Hamel Basis ...
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43 views

Infinite basis concerning the polynomials

I read an answer in the post that $\{1,x,x^2,\cdots\}$ forms a basis for $\mathbb{C}[x]$. I want to extend it a bit more general: Is it true that if $p_0,p_1,\dots$ are any polynomials from $\mathbb{...
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Finding a vector given a basis vector and coordinate vector

I was given the following question: Find the vector $x$ determined by the given coordinate vector $[x]_B$ and the given basis B. $B=\left(\left[\begin{matrix}1\\1\\\end{matrix}\right],\left[\begin{...
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56 views

Taft-Hopf Algebra has dimension $N^2$?

Definition of the Taft-Hopf Algebra Let $k$ be a field. Let be $N$ a positive integer such that there exists a primitive $N$-th root of unitiy $\zeta$ over $k$. Denote by $(H, \mu, 1_H)$ the unital, ...
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Another approach for dimension of a vector space

I am preparing a lecture note for a primary course on Vector Spaces and I am developing basis and linearly independent sets. There I took the following path: Define the linear span $L(S)$ of a subset ...
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If $x$ and $y$ are linearly independent, there is a separating linear functional

Let $E$ be a $\mathbb R$-vector space and $x,y\in E$ be linearly independent. How can we show that there is a $\varphi\in E^\ast$ with $\varphi(x)=0$ and $\varphi(y)=1$? I guess we need to construct $...
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Baire Category Theorem in infinite- dimensional space

Let $X$ be an infinite-dimensional Banach space. Suppose that $ M$ is an infinite- dimensional subspace of $X$ that has a countable Hamel basis. I want to show that $ M $ is a meager subset of $ ...
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39 views

Uncountable Hamel basis of Banach space

I came across this problem: Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.. The proof is absolutely correct however I was wondering if this is true ...
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What's the dimension of $K^S$ for arbitrary $S$?

Let $K$ be a field, $S$ an arbitrary set, and $K^S$ denote the vector space of functions from $S$ to $K$. What is the dimension of this space? By dimension, I mean the cardinality of a Hamel basis. ...
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Show that a linear functional $f$ on a vector space $X$ is uniquely determined b its values on a Hamel basis for $X.$ [closed]

Show that a linear functional $f$ on a vector space $X$ is uniquely determined by its values on a Hamel basis for $X$. I need to prove this.
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Are those 2 questions asking about the same thing?

Here is the first question: Let $X$ and $Y \neq \{0\}$ be normed spaces, where $dim X = \infty.$ Show that there is at least one unbounded linear operator $T: X \rightarrow Y.$ (Use a Hamel basis.) ...
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Define continuous functions mapping real numbers between [0, 1] to basis elements of C^n

I am looking at some old lecture notes on Linear Algebra, in which the following exercise is given: My first thought was to simply define $x_{i}(t) = (1-t)u_{i} + tv_{i}$, but this would fail if $(1-...
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Two basis for a vector Space $V$ has the same coordinates. Does that follow both basis are identical?

Let $\beta_{1} = \{v_1,v_2,\ldots,v_n\}$ and $\beta_{2} = \{u_1,u_2,\ldots,u_n\}$ be two bases for some vector space $V$. If the coordinates for every vector $x\in V$ are identical with respect to ...
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Showing that this vector space is infinite dimensional

Let $$V:=\bigcup_{n=1}^{\infty}\left\{(x_i)_{i=1}^{\infty}:x_i\in\mathbb{C},\space{}x_i=0\space{}\forall{}\space{}i>n\right\}.$$ I am trying to show that $V$ is infinite dimensional and that its ...
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On linear combinations and bases in $\ell^\infty$

Just like in these two questions: 1, 2, I was also struggling to understand bases in $\ell^\infty$ in a constructive way, which I understood was not possible. However, in order to get a better ...
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Let's say there are 2 basis spanning a subspace. One of the basis is a scalar multiple of the other basis?

So, can the vector that is a scalar multiple of one of the basis out of the 2 basis vectors spanning the subspace? I dont know how to paraphrase my question. In case, you dont get my question, I ...
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Polynomial Sub-Spaces [closed]

Let $ V =\mathbb{R}_2[X]$ and $U = \{p\in V \mid p(-1) = p(2)\}$ Show that $U$ is a Linear Subspace of $V$ , and find a basis for $U$. Complete the basis of $U$ to a basis of $V$. I understand ...
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linear combination, span, independence and bases for infinite dimensional vector spaces.

I've only recently started studying linear algebra using some lecture notes by Evan Dummit (https://math.la.asu.edu/~dummit/docs/linalgprac_2_vector_spaces.pdf). After defining vector spaces, the ...
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Give a basis for $S = \{M \in \mathcal{M}_3(\mathbb{K})\ |\ \text{Tr}(M) = 0 \wedge \sum_{j=1}^3 M_{i,j} = 0\}$

I'm looking for a basis of the vector space formed by the set $$S = \left\{M \in \mathcal{M}_3(\mathbb{K})\ |\ \text{Tr}(M) = 0 \wedge \forall i \in [\![1, 3]\!]: \sum_{j=1}^3 M_{i,j} = 0\right\}$$ ...
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About Cauchy's Functional Equation

It's known that the solutions to the Cauchy's Functional Equation: $$f(x+y) = f(x) + f(y)$$ are of the form $f(x) = cx$, where $c$ is a constant. I've read that this situation takes place when the ...
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215 views

Applications of real numbers being a vector space over the rational numbers

What applications are there of the fact that the real numbers form a vector space over the rational numbers? Vector spaces over the rational numbers appear to have uses in number theory. The ...
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Is Hamel Basis necessarily uncountable?

Let $X$ be a (real or complex) infinite dimensional vector space. (Not Normed or Banach one). Is every Hamel Basis for $X$ necessarily uncountable ?
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Can this be a Schauder basis of $\mathbb{R}[[x]]$

A few hours ago, I asked a question about using Taylor expansion of two analytic functions on $\mathbb{R}$ to determine whether these two functions are linearly independent. Basically I was trying to ...
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Linear independent vectors - Span - Basis

Let \begin{equation*}x_1:=\begin{pmatrix}1 \\ -1 \\ 1 \\ -1\end{pmatrix}, \ x_2:=\begin{pmatrix}2 \\ 0 \\ 3 \\ -1\end{pmatrix}, \ x_3:=\begin{pmatrix}-2 \\ 1 \\ 0 \\ 3\end{pmatrix}, \ y:=\begin{...
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Doubt in the Hamel Basis of infinite Dimension Vector space

I know that by Zorn's Lemma we can prove that every vector space has a Hamel Basis. Where Hamel Basis means a maximal Linearly independent set. My question is, if this is the Finite Dimension case, ...
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Determine all $f:\Bbb R\to \Bbb R$ such that $f\big(a-3f(b)\big)=f\left(a+f(b)+b^3\right)+f\left(4f(b)+b^3\right)+1$ for every $a,b\in\Bbb R$.

Im struggling with this functional equation: Determine all $f: \Bbb R \to \Bbb R$ such that $$f\big(a-3f(b)\big)=f\left(a+f(b)+b^3\right)+f\left(4f(b)+b^3\right)+1$$ for all $a,b\in\Bbb R$. Clearly ...
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Element in the linear span?

Let $H$ be a Hilbert space and $\{x_i\}_{i\in I}$ a countably infinite set. Denote $S:=\overline{\operatorname{span}\{x_i\}_{i\in I}}$. If $\sum\limits_{i\in I}|{a_i}|^2<\infty$ is then $$\sum\...
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Basis of vector space and finite linear combination of any element

Kreyszig1989 (quote unchanged inclusive exclamation mark): More generally, if $X$ is any vector space, not necessarily finite dimensional, and $B$ is a linearly independent subset of $X$ which ...
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Problem regarding Hamel basis

I'm doing a problem for a homework assignment and the problem is as follows: Let $\{e_{\alpha}\}_{\alpha\in I}$ denote a Hamel basis for a vector space $X$. (i) Let $T:X\rightarrow X$ be a bijective ...
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Why a Hamel basis?

I think I understand what both Hamel basis and Schauder basis mean. But to me, the Schauder basis makes more sense intuitively than the Hamel basis does. For example, Fourier series, Bases for Hilbert ...
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Doubt about the notion of “unique representation of a vector $v$” given by Direct Sums and Basis sets

$\newcommand{\Span}{\mathrm {span}}$Consider only finite dimensional vector spaces with a "ordinary" field (like $\mathbb{R}$ or $\mathbb{C}$). Moreover I appreciate a discussion based only on ...
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Name for a multiplicative analogue of a Hamel basis?

Let $\mathfrak B$ be a Hamel basis for $\mathbb R$ over $\mathbb Q$. Then the set $\mathfrak{M} = \left\{ e^b | b \in \mathfrak{B} \right\}$ has the property that any $r\in\mathbb R^+$ can be ...
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Generation of a basis of an infinite dimensional space using a finite number of vectors

Suppose the set $\mathscr{H} = \{\phi_i\}_{i=1}^{\infty}$ is a complete orthogonal set (of functions) with a given inner product operation. Let $\mathscr{H}_N = \{\phi_i\}_{i=1}^{N}$, i.e. the first $...
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Definition for Basis of a Subspace

As I gradually work through the fundamentals of linear algebra, I have often found myself struggling with boiling down the concepts that I have learned to a concise, all encompassing definition which ...
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What are bases, really?

I'm taking a course in Linear Algebra right now, and am having a hard time wrapping my head around bases, especially since my prof didn't really explain them fully. I would really appreciate any ...
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172 views

a linear transformation must be injective, supposing it is surjective and also has a special property with respect to spanning sets

Proposition: Let $(V,F)$ and $(W,F)$ be non-zero vector spaces, and $T$ be a surjective linear map of $V$ onto $W$. Assume that property $(1)$ holds: for any subset S of V, we have that "TS ...