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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Vector Field Decomposition into Electrostatic Functions

I have a question that is motivated by an engineering problem: Say I am given a general 2d vector field $F:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. I want to find a configuration of point charges in $\...
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The set of vectors {a, b, c} is a basis of R³. Determine if another set, {3a-2b, a+4b+5c, a-2c}, is also a basis of R³.

Since the set of vectors $\{\vec{a},\vec{b},\vec{c}\}$ are a basis of $\mathbb{R^3}$, we can understand that they are linearly independent. This means that: $\alpha_1\cdotp\vec{a}+\alpha_2\cdotp\vec{...
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Is a Hilbert space determined by its algebraic dimension? [duplicate]

Every Hilbert space is determined up to a unitary isomorphism by the cardinality of its orthonormal basis (which exists by Gram-Schmidt). My question: Let $\mathbb{K}$ be the field either $\mathbb{R}$...
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Basis of a polynomial ring over the field of Rationals

I need help checking the validity of the method of my proof and completing it. We observe the polynomial ring $\mathbb{Q}[x] $ as a vector space over the field $\mathbb{Q}$ it is easy to see from the ...
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Boundness of the operator when Hamel basis vectors are eigenvectors

Let $X$ be infinite-dimensional Banach space, $\{e_j\}_{j\in J}$ be normalized Hamel basis for it. Suppose for each $j\in J$ we choose a number $\lambda_j\in\mathbb{R}$. Then we define a linear map $$...
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Largest and smallest norm on a vector space given a Hamel basis should be normalised in that norm

I don't know whether the answer to this question will help answering this question. Let $V$ be an infinite-dimensional vector space and $A \subset V$ a Hamel basis in $V$. What is the largest (and ...
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Why isn't this a counterexample of Banach-Steinhaus theorem?

The theorem from Wikipedia is as follows Let $X$ be Banach space, $Y$ be a normed space and $F$ be family of linear bounded operators $f:X \to Y$ such that $\forall x \in X \sup_{f \in F} \|f(x)\|_Y &...
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What is a Hamel basis?

According to Mathworld, a Hamel basis is a basis for $\mathbb R$ considered as a vector space over $\mathbb Q$. According to Wikipedia, the term is used in the context of infinite-dimensional vector ...
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How to Find the Basis for a Specific Vector Space? [closed]

I have the following homework problem: In Exercises 1-4, $W$ is a subspace of the vector space $V$ of all $(2 \times 2)$ matrices. A matrix $A$ in $W$ is written as A = $$ \begin{bmatrix} a & b \\...
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Basis for a subspace of matrix $2\times2 $with sum 0 entries.

Consider the subspace of matrices of the form $\left(\begin{matrix} a & b\\ c & d \end{matrix}\right)$ such that $a+b+c+d=0$ I have to find a basis. I was thinking for example of letting $3$ ...
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What am I missing in this paper: ``Infinite dimensional Banach spaces must have uncountable basis—an elementary proof"?

In the paper (or refer to this answer to view the full proof of this result): Tsing N.K. [1984]. Infinite dimensional Banach spaces must have uncountable basis—an elementary proof. Amer. Math. ...
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Proving $d\varphi^{\mu_1} \wedge d\varphi^{\mu_2}... \wedge d\varphi^{\mu_k}$ is basis for space of $k$-forms

Let $X$ be a smooth $n$-manifold with a local chart $(U,\varphi)$. Taking the wedge product of the chart induced covariant basis $\{d\varphi^\mu\}$: \begin{align} \{d\varphi^{\mu_1} \wedge d\varphi^{\...
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setting up for basis of vector

I don't know how to get started with this Find a basis for the subspace $V$ of $\mathbb{R}^4$ consisting of all vectors of the form $(a,b,c,a+b+c)$, and state its dimension. Another way or writing ...
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Can this theorem be generalized to infinite dimensional vector spaces?

I am reading some basic linear algebra theory, and I came across this theorem. Let $V, V'$ be vector spaces and $\ \dim V \lt \infty$ Then for every basis $e_1, e_2, \dots, e_n$ of $V$ and any $n$ ...
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Direct proof from AC that every vector space has a basis?

Let AC be the axiom of choice and VB be the proposition that every vector space has a basis. What's the "most direct" possible proof that AC$\implies$VB (in ZF)? I know it's possible to ...
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Suppose $w$ is any vector in $V$, then, for some choice of sign $\pm$, $\{v_1\pm w, v_2,\cdots,v_n\}$ is a basis for $V$.

Question: Let $\{v_1,\cdots,v_n\}$ be a basis for vector space $V$ over $\mathbb{R}$. Suppose $w$ is any vector in $V$, then, for some choice of sign $\pm$, $\{v_1\pm w, v_2,\cdots,v_n\}$ is a basis ...
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How to show that the reciprocal set is also a basis of $\mathbb{R}^3$

If three vectors (say $\vec v_1$, $\vec v_2$ and $\vec v_3$) are a basis of $\mathbb{R}^3$, their reciprocal vectors ($\vec u_1$, $\vec u_2$ and $\vec u_3$, defined by $\vec{u}_i \cdot \vec{v}_j = \...
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Basis of a completion of a vector space

Let $V$ be an infinite dimensional vector space with basis $\{e_i\}_{i \in \mathbb{N}}$ so that every element can be written as a finite linear combination of the $e_i$'s. Let $V: = V_0 \supseteq V_1 \...
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Finding the basis of ker(T) and im(T)

Question : Let $P_2$ be the space of polynomials of degree less or equal to 2. a.) Write down a basis $B$ for $P_2$. b.) Define the linear transformation on $P_2$ by $T(f)(x) = f^{\prime}(x), $ the ...
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For any vector space with a basis, does there exist a choice of inner product w.r.t. which the basis is orthonormal?

Suppose you have a vector space $V$ (without an inner product) with a basis $B = \{ \vec{v}_i \}$. Does there always exist a choice of inner product $I$ with which you can endow $V$ that will render ...
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Is the span closed?

Let X be a normed space. Let $v_1, v_2... v_n$ be vectors in X. When is $Span({v_1, v_2, ... v_n})$ closed? This question is motivated by a question which I had on a problem sheet (this wasn't the ...
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Clarification Regarding Banach space and Baire Category Theorem

New to Baire Category. From a remark: If we suppose that $Y$ is an infinite dimensional subspace of a Banach space $X$, and $Y$ has a countable (Hamel) basis, then one can show that $Y$ is first ...
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Hamel Basis for Homeo$^+(\mathbb{R}^2)$

In the 50's through 70's there was a lot of research into the group of orientation-preserving homeomorphisms of the plane, denoted as Homeo$^+(\mathbb{R}^2)$ (in the compact-open topology, which in ...
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Finding basis and explain why the basis have found is a basis

Let u = (1, 2, 3, 4) and v = (4, 3, 2, 1) be two vectors in $R^4$. These vectors define the subset of $R^4$ $V = \{x \in R^4 | u \bullet x = 0$ and $v \bullet x = 0\}$ Here $u \bullet x$ denotes ...
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A nonmeasurable subset of a Hamel Basis

The following is an exercise from Bruckner's Real Analysis: Let $H$ be a Hamel basis and $H_0$ a nonempty finite or countable subset of $H$ . Show that the set of rational linear combinations of ...
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Tying Span, Basis, Linear Independence, Dimension Together [closed]

I'm going through Linear Algebra by Friedberg et al. and I'm having trouble with the buildup to the main ideas regarding span, linear independence, basis, and dimension. I'll detail what I've taken ...
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If $\beta$ is a basis of $V$ then does that mean that $V=span{\beta}$?

I am confused with the definition of 'basis'. A basis $\beta$ for a vector space $V$ is a linearly independent subset of $V$ that generates $V$. And span($\beta$) is the set consisting of all linear ...
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Is the direct sum decomposition of formal power series provable without choice?

If we view the ring of formal power series $F[[x]]$ as a vector space over $F$, and we view the polynomial ring $F[x]$ as a subspace of $F[[x]]$, then the axioms of choice implies that $$F[[x]]=F[x]\...
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Basis of $V=\{a\cdot(1,2,3)^T\}$

I've a vector space $V=\left\{a\left(\begin{array} {l}1 \\ 2 \\ 3\end{array}\right)\right\}$ $a$ is any real number. Can I choose it's basis as $\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$. ...
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Prove that if there exists an infinite linearly independent subset E, of a vector space V, then V is infinite dimensional.

Def: A Hamel Basis for V is a linearly independent subset that spans V. Def: If V has a finite Hamel Basis, F, then the dimension of V is equal to the number of elements in F, if not V is infinite ...
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Checking if given set forms basis for $P_3(\mathbb{R})$

The question says whether the polynomials $x^3+2x-4,\ x^3+x^2-3x+1, $and $x^3+5$ generate $P_3(\mathbb{R})$? I did the question in a sneaky way as follows. We already know $\{1,x,x^2,x^3\}$ forms the ...
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Why is it important that every (infinite) dimensional vector space has a (hamel) basis?

An argument often used in favor of the axiom of choice is that it is equivalent to every infinite dimensional vector space having a hamel basis. However the article on wikipedia says that those basis ...
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Hamel basis for $\mathbb{R}_{\geq 0}$ as a vector space over $\mathbb{F}_2$

Let $\mathbb{F}_2 = \{0,1\}$ be the field with two elements. The naturals $\mathbb{N} = \{0,1,2,3,...\}$ (with vector addition given by binary bitwise XOR) form a vector space over $\mathbb{F}_2$, ...
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If $\mathcal V:=\{\vec v_i\}$ and $\mathcal U:=\{\vec u_i\}$ are bases that is $\mathcal V_i:=\{\vec v_1,...,\vec u_i,...\vec v_n\}$ a base?

So let be $V$ a vector space of finite dimension and let be $\mathcal V:=\{\vec v_i:i=1,...,n\}$ and $\mathcal U:=\{\vec u_i:i=1,...,n\}$ two different bases so that there exist $i=1,...,n$ such that $...
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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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dimension of infinite dimensional vector space

Let $V$ be a vector space over a field $\mathbb{K}$ (either $\mathbb{C}$ or $\mathbb{R}$) that has an infinite linearly independent subset. Prove that if $B$ and $B'$ are two bases for $V,$ then $B$ ...
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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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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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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{...