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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Why in a finite dimensional space every orthonormal basis is basis

Why in a finite dimensional space every orthonormal basis is basis i know in infinite dimensional space every basis is orthonormal basis but converse is not true ( for example $l^2$ ) but in finite ...
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2answers
34 views

Open set minus closed set with empty interior

Let $X$ be a separable infinite-dimensional Banach space, $U$ be non-empty and open in $X$ and $E$ be finite-dimensional linear subspace of $X$. I would like to know if there is a non-empty open ...
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1answer
37 views

Eigendecomposition of rank-deficient rank-$1$ update

I have a matrix of the form $C = I - a a^T$. In this particular case, $a$ is a constant vector with value $1/\sqrt{N}$, where $N$ is the rank of $I$ (number of columns). For $I = eye(5)$, $N$ will be ...
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1answer
41 views

Every basis in a finite dimensional Hilbert space is a Riesz Basis

The goal is to prove that every basis in a finite-dimensional Hilbert space is a Riesz basis, i.e., there exist constants $A>0$ and $B>0$ for the basis $\{x_k\}$ such that: $$ A \sum_n |a[n]|^2 \...
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2answers
22 views

Find orthogonal basis of a space

I am trying to find an orthogonal basis of a space W defined by vectors. $ W=[(0,1,0,1),(1,1,0,1),(0,0,0,1)] $ . How would I achieve so? I have no idea how to begin. In my textbook there is a hing to ...
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If $U$ satisfy $dim\, U=0$ then what U could be? [closed]

Is it just an empty space? is the group $\left \{ 0 \right \}$ is linearly independent?
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30 views

In every nonzero vector space, each element has a unique representation as the linear combination of finitely many elements.

I can't understand this statement from Kreyszig's Introductory Functional Analysis with Applications. How is it possible to have every infinite dimensional vector space's entire set of elements can ...
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1answer
40 views

Basis of $\mathbb{Q}(\sqrt[3]{2},\omega)$ over $\mathbb{Q}$

I found this resource by Dr. Keith Conrad showing how to compute the basis of $\mathbb{Q}(\sqrt[3]{2},\omega)$ over $\mathbb{Q}$ (where $\omega$ denotes the third primitive root of unity): Keith ...
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2answers
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Every basis of a subspace has the same cardinality

I'm stuck on proving the following theorem: "Let $B = \{u_1,u_2, . . . ,u_m\}$ and $B' = \{v_1, v_2, . . . , v_k\}$ be bases for a non-zero subspace $S$ of $\mathbb R^n$. Then $m = k$. That is, any ...
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4answers
51 views

Define dimension without referring to bases

The dimension of a vector space is the common cardinality of all bases. I would like to define it in a way that does not refer to bases. I only care about finite dimensional vector spaces. First, I ...
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1answer
32 views

Will the span of vectors {(1,0,0,0,0),(0,1,0,0,0)} form a subspace/basis in the set of fields of dimension 5?

I have checked a calculator website which checks if a set of vectors is a basis and the one that I put in the title is not. (http://www.mathforyou.net/en/online/vectors/basis/) By the subspace ...
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Does the existence of a linear functional on $C_0^\infty(\mathbb R)$ which is not a distribution require the axiom of choice?

Consider $C_0^\infty(\mathbb R)$ as a real vector space. By choice, we can take a Hamel basis $\{e_\alpha\}_\alpha$ for $C_0^\infty(\mathbb R)$ such that every $f\in \mathbb C_0^\infty(\mathbb R)$ can ...
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Maximal linear subspace in a linear space

$V \ \ is\ \ a\ \ linear\ \ space\ \ ,\ \ \ \ \ \varphi_1, \varphi_2, ... ,\varphi_n \ \ are\ \ linear\ \ independence\ \ ,\ \ and\ \ \Psi \in V^{'}$ Now $$ \bigcap_{i=1}^n ker\ \varphi_i\ \ \subset ...
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23 views

Does the statement “Every short exact sequence of vector spaces splits” imply the axiom of choice? [duplicate]

Using the axiom of choice, or more directly, the statement that every linearly independent set of vectors in a vector space may be extended to a basis, it is easy to prove that every short exact ...
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1answer
63 views

Is it true? $\dim(U+W)=\dim(U\cap W)+1$ implies $U\subseteq W$

I'm curious whether or not this statement is true. If $U,W$ are finite-dimensional space satisfying $\dim(U+W)=\dim(U\cap W)+1,$ then $U\subseteq W.$ Any help would be appreciated! Thank you
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Existence of orthogonal Hamel basis for infinite dimensional vector space.

Consider a $\mathbb{F}$-vector space $V$ that is infinite dimensional and equip it with an inner product (i.e. it is a Pre-Hilbert space). We know that $V$ has a basis set, say $S$, such that for any $...
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1answer
35 views

Why is $ (a, b) = 0 $ for distinct $ a, b \in A $, the Hamel basis of vector space $ \mathbb{R} $ over $ \mathbb{Q} $?

Let us consider the set of real numbers $ \mathbb{R} $ as a vector space over the set of rationals $ \mathbb{Q} $. We know that this vector space has a basis known as the Hamel basis. Let the Hamel ...
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0answers
52 views

How to find a basis of $\Bbb R(t)$ over $\Bbb R$? [duplicate]

I am reading field theory. I dont understand the following fact: What will be a basis of $\Bbb R(t)=\{\frac{f(t)}{g(t)}:f(t),g(t)\in \Bbb R[t]\}$. I know that $\Bbb R(t)$ is a field and hence a ...
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3answers
53 views

Find a linear map knowing its image and kernel

So..I have to find any linear map $f: \mathbb{R}^4 \rightarrow \mathbb{R}^3$ that has a kernel and and an image with the following basis: $$\ker(f)=\operatorname{Span}\{(-1,0,0,1),(1,3,2,0)\}.$$ $$\...
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1answer
40 views

What is the general method of expressing a basis in terms of another basis of a vector space?

I want to understand basis of vector spaces more clearly. I know that the basis of a vector space is a set of linearly independent elements that span that vector space. I know that an element in a ...
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1answer
37 views

how to prove that a linear transformation maps every basis to the basis of the coordinate space

This is my homework's problem: Let $T$ be a linear transformation that maps $V$ to $W$. Prove that this transformation maps every basis of $V$ to the basis of $W$. I think I knew the answer. I want ...
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2answers
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$\{x_1,…,x_n\}$ a basis of $V_1$; SHOW THAT there is a unique linear transformation $f:V_1\to V_2$ such that $f(x_1)=y_i$

The following question is given in the book Linear Algebra by A.R. Rao and P. Bhimasankaram: Let $V_1$ and $V_2$ be vector spaces over $F$ and let $\{x_1,...,x_n\}$ be a basis of $V_1$. Then, for ...
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1answer
32 views

Let $v \in \operatorname{Im}(p)$. Compute $p(v)$.

Let $B = (1, X, X^2)$ be an ordered basis for $\Bbb R_2[X]$ and $p ∈ \mathcal{L}\big(\Bbb R_2[X]\big)$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)...
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0answers
67 views

An incomplete inner product space with an orthonormal basis that is not a Hamel basis

Given any infinite dimensional Hilbert space $X$, the linear span, $Y$, of any orthogonal basis for $X$ is an incomplete inner product subspace of $X$. The orthogonal basis for $X$ is an orthogonal ...
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1answer
52 views

Three space property

I want to show that Finite dimensionality is a three space property. Let $X$ be a normed linear space and let $Y$ be a closed subspace of $X$. If $Y$ and $X/Y$ are finite dimensional spaces, then I ...
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1answer
33 views

Finding a basis for a eigenspace

I have the linear transformation $L_1: V \rightarrow V$ (where $V = \mathbb{C}[x,y]$ is the vector space of polynomials with complex coefficients) defined as: $L_1(f(x,y)) = \frac{f(x,y)+f(y,x)}{2}...
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1answer
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Finding a Basis for the Vector space of sequences of the form $u_{n+1} = u _{n-1} + u_n$?

The question asks to show that the set of real sequences $u_n$ satisfying the recurrence $u_{n+1} = u_n + u_{n-1}$ is a subspace of the space of all real sequences and then to find its basis. To ...
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5answers
101 views

Showing $\mathbb{R}^2$ has a Hamel basis using Zorn's lemma?

To get some intuition for Zorn's lemma I want to use it explicitly in the proof of the following theorem in the case when $X = \mathbb{R}^2$: Every vector space $X \neq \{ 0\}$ has a Hamel basis. ...
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1answer
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Finding a Basis of a Vector Space - Coefficients

I understand that first you have to prove that it's linearly independent, but wouldn't a coefficient of 0 always work? Ex: $a_1(u+v+w) + a_2(-2u+v-w) = 0$ For any vector, wouldn't $a_1 = a_2 = 0$ ...
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1answer
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How to find the intersection of $W$ and $Z$? [duplicate]

Subspaces$W$ and $Z$ of $\mathbb R^4$ are generated by $\{(1,1,0,-1),(1,2,3,0),(2,3,3,-1)\}$ and $\{(1,1,0,-1),(1,2,3,4),(0,1,3,5)\}$, respesctively. Find a basis for $W$$\cap$$Z$. I already ...
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1answer
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Does the set $\{2^q : q\in \mathbb{Q}, 0\le q < 1\}$ form a linearly independent set over the field $\mathbb{Q}$ [closed]

My intuition says yes, but I'm having a lot of difficulty proving it.
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2answers
56 views

$U=\{ p\in P_{4}\left( \mathbb{R} \right) : p''\left( 6\right) =0\}$ Find a basis for $U$

Find a basis for $U$, where $$U=\{ p\in P_{4}\left( \mathbb{R} \right) : p''\left( 6\right) =0\}.$$ This question is 2.C 5 of Linear Algebra Done right. I would like to know how this person got ...
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3answers
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Constructing a new basis

Let $B=\{v_1,\dots,v_n\}$ be a basis for a vector space $V$. Show that $A=\{v+v_1,\dots,v+v_n\}$, where $v=\sum_{i=1}^n a_iv_i$ is a basis for $V$ if and only if $\sum_{i=1}^n a_i\neq -1$. Of course ...
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3answers
277 views

Can a Hamel basis for an infinite-dimensional Hilbert space be orthonormal?

For a finite-dimensional Hilbert space, any orthonormal basis is trivially a Hamel basis (because there's only one natural notion of "basis" in finite dimensions). But for an infinite-dimensional ...
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1answer
76 views

I wanna know about a theorem of linear algebra

Is there some theorem that says that if the number of vectors of a base $S$ is less than the dimension of a vector space $V$ than it cannot span the vector space?
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1answer
40 views

Benefits of choosing a Hamel bases for $L^p$ including a specific linearly independent subset

According to the book I am studying (Royden & Fitzpatrick): We can infer from Zorn's Lemma that every linear space possesses a Hamel basis. A Hamel Basis is defined as a subset $\mathcal{B} $ ...
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3answers
152 views

We have $n$ real numbers around the circle and among any consecutive 3 one is AM of the other two. Then all the numbers are the same or $3\mid n$.

There are $n$ real numbers around the circle and among any consecutive 3 one is arithmetic mean of the other two. Prove that all the numbers are the same or $3\mid n$. Hint was to use a linear ...
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1answer
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null,row and column space can have different bases

When solving examples that ask me to find a basis for the null, row or columnspace for a matrix I often get a different basis than what I find in the answer sheet. I know you can come up with many ...
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2answers
87 views

Dual Space Subset Existence Proof.

Let $V$ be a finite-dimensional vector space. Let $V^*$ be the dual space of $V$. Choose a basis $ \mathcal{B} = \{\textbf{e}_1,\dots,\textbf{e}_n\}$ for $V$. I would like to prove that there exist $\...
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5answers
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Are there vector spaces with uncountable basis? [duplicate]

Are there vector spaces with uncountable basis ? I was thinking about something as $L^1(\mathbb R)$. A could imagine that $\varphi_x:\mathbb R\to \mathbb R$ defined as $\delta_x(y)=1$ if $y=0$ and $0$ ...
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4answers
120 views

Using basis $e=[x^3,x^2,x,1]$ instead of $e=[1,x,x^2,x^3]$

So on an exam I've got zero points on the question (and sub-questions) to find matrix of linear operator $L:\Bbb{R}^4[x]\to \Bbb{R}^4[x]$ given by $L(p(x)) = p(x)+xp(2)$ with respect to canonical ...
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1answer
78 views

Basis on a vector space with a filtration

Let $V$ be a vector space over some field $F$. Consider a filtration of $V$ by subspaces $V_k$ ($k \in \mathbb{Z}$) such that $V_k \subseteq V_l$ for all $k \leq l$. That is, we have a filtration $$...
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2answers
60 views

Hamel Basis. Consequences?

I am not sure to have the concept clear. Where is my reasoning flaw? According to the proven existence of a Hamel Basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, then it is possible to ...
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1answer
57 views

Counterexample(?) of the theorem An Orthonormal basis of a vector space X is a Hamel basis if and only if X is finite dimensional.

I found somewhere the following theorem. An Orthonormal basis of a vector space X is a Hamel basis if and only if X is finite dimensional. For Hilbert spaces it is quite easy to prove but for pre-...
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3answers
72 views

Find a Basis for $W=\{p(x)\in V: p(1)=p'(1)=0\}$

Let $V=\mathbb{P_4}$ and $W=\{p(x)\in V: p(1)=p'(1)=0\}$. Assuming that $W$ is a subspace of $V$, find a basis for $W$ and thereby determine the dimension of $W$. I think that $\dim(W)=3$ as there ...
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2answers
1k views

Show that a vector can be expressed as a linear combination of the vectors that form a basis for its vector space in exactly one way

Show that if S = {$v_1$, ... , $v_n$} is a basis for a vector space V then each vector v $\in$ V can be expressed as v = $k_1v_1$ + $k_2v_2$ + ... + $k_nv_n$ (where $k_i \in R$ for i = 1, ... , n) ...
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3answers
324 views

Showing Bernstein polynomial is a basis

Hello I want to show that the Bernstein polynomial $$B_{n,k}=\binom{n}{k}x^k(1-x)^{n-k}\,$$ is a basis. For linear independece I got a hint from my teacher to expand the binom $(1-x)^{n-k}$ This way I ...
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1answer
95 views

What does exterior algebra actually mean?

This question may be too basic and even silly, but I am new to exterior algebra and reading Wikipedia. Given $e_1, e_2,\cdots, e_n$ is a standard basis for a vector space $V$, what does $e_1\wedge ...
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2answers
188 views

Relation between identity transformation and transformation matrix

Suppose $T: V \to V$ is the identity transformation. If $B$ is a basis of V, then the matrix representation of $[T]^B_B = [I_n]$. Let's say C is also a basis of V, then it is clear that $[T]^B_C \...
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2answers
183 views

An orthogonal basis of a Hilbert space is Schauder?

I read that an orthogonal (Hamel) basis $(e_i)$ of a Hilbert space is always a Schauder basis. I can see why; if $\sum \alpha_i e_i=0$ then taking the inner product with each $e_i$ gives $\alpha_i =0$....