So the real numbers for a vector space over the rational numbers. The dimension is infinite. Is it countably infinite or uncountably infinite?
2025-01-13 05:34:19.1736746459
What is the basis of the vector space of real numbers over rational numbers?
2.6k Views Asked by aksd https://math.techqa.club/user/aksd/detail At
1
There are 1 best solutions below
Related Questions in VECTOR-SPACES
- Show that $f(0)=f(-1)$ is a subspace
- $a*x=0$ implies $x=0$ for all scalars $a$ and vectors $x$.
- Vector space simple propertie
- Is the row space of a matrix (order n by m, m < n) of full column rank equal to $\mathbb{R}^m$?
- orthonormal basis question - linear algebra
- Expressing a Vector in a new Basis
- Linear Algebra: Let $w=[1,2,3]_{L_1}$. Find the coordinates of w with respect to $L$ directly and by using $P^{-1}$
- Direct sum counterexample
- Prove $(\sum\limits_k a_k b_k)^2 \leq \sum\limits_k b_k a_k^2 \sum\limits_k b_k$ using Cauchy Schwarz
- Proving that $\dim(U ⊕ W ) = \dim U + \dim W$
Related Questions in REAL-NUMBERS
- The integer part of $x+1/2$ expressed in terms of integer parts of $2x$ and $x$
- Prove implication all Cauchy sequences have a limit $\to$ all monotone increasing bounded above sequences converge$
- definition of rational powers of real numbers
- on the lexicographic order on $\mathbb{C}$
- Finding percentage increase
- Neighbors of Irrational Numbers on Real Number Line
- Correct notation for "for all positive real $c$"
- Given two set of real number the intersection between this two set is an interval?
- Let $b\in\mathbb{R}$ and $b>1$. Show that for all $r\in\mathbb{R}\exists n\in\mathbb{N}$ such that $r<b^n$
- Real numbers for beginners
Related Questions in HAMEL-BASIS
- What is the basis of the vector space of real numbers over rational numbers?
- What's an example of a vector space that doesn't have a basis if we don't accept Choice?
- Proving that basis always exists and is not unique
- Why isn't every Hamel basis a Schauder basis?
- Hamel Bases: Cardinality?
- Element in the linear span?
- 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 $a,b\in\Bbb R$.
- Doubt in the Hamel Basis of infinite Dimension Vector space
- Linear independent vectors - Span - Basis
- Is Hamel Basis necessarily uncountable?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity
It is going to have the same cardinality as $\mathbb{R}$, so uncountable.