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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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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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. . 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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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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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 ...
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{...