Intereting Posts

Why continuum hypothesis implies the unique hyperreal system, ${}^{\ast}{\Bbb R}$?
What is a composition of two binary relations geometrically?
Compute the characteristic equation 3×3 matrix
Block Diagonal Matrix Diagonalizable
Determine whether $F(x)= 5x+10$ is $O(x^2)$
Implicit Differentiation Help
Assistance with proof of $(AB)^T=B^T A^T$
Primes of ramification index 1 with inseparable residue field extension
Does the series $\sum_{n=1}^{\infty}|x|^\sqrt n$ converge pointwise? If it then what would be the sum?
Dirac delta integral with $\delta(\infty) \cdot e^{\infty}$
Pointwise but not Uniformly Convergent
number of pairs formed from $2n$ people sitting in a circle
Strangest Notation?
Characterization of hierarchically clustered graphs
How to prove the distributive property without using truth tables?

I’m attempting to come up with a first order language $L$ that is able to describe vector spaces over fields. I came up with a few sets of nonlogical symbols.

$Rs_L=\{Scal,Vec\}$ where $Scal,Vec$ are unary relation symbols which I interpret to be true or false depending on whether $x$ is a scalar or vector.

Also, $Fs_L=\{+,\cdot,\}$ which are the usual addition and multiplication for both vectors and scalars.

- How do you rearrange equations with dot products in them?
- L : $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is a linear mapping, linear independence of $L$ mapped onto a set of vectors.
- Adjoint matrix eigenvalues and eigenvectors
- Does the given matrix have a square root?
- A normal matrix with real eigenvalues is Hermitian
- System of Linear Equations - how many solutions?

For constant symbols, $Cs_L=\{0,1,0_V\}$, where $0$ and $1$ are regular field scalars for the additive and multiplicative identities, respectively, and $0_V$ is the additive identity for vectors.

Using these, I was able to formulate axioms for scalar multiplication on a given structure, and I define addition between a scalar and a vector to simply always give the 0 vector and define multiplication between two vectors to always be the 0 vector. This gives a theory such that a structure models it if and only if it is a vector space in the regular algebraic sense. It is easy to see that all linear transformations in the algebraic sense are included in the class of homomorphisms (in the logical sense), but I am having difficulty proving they are the only ones in this class.

I suppose I want to show that for any $L$-homomorphism $T$, $T(av+w)=aTv+Tw$ for $a$ a scalar and $v,w$ vectors. Since $T$ is a homomorphism, I see that $T(av+w)=Tav+Tw=TaTv+Tw$, so it would be enough to show that all homomorphisms act as the identity on all scalars? Is there a way to show this, or an extra axiom to include?

- Triangular matrices proof
- How to show if $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$?
- Projecting a nonnegative vector onto the simplex
- Does an injective endomorphism of a finitely-generated free R-module have nonzero determinant?
- Physical meaning of the null space of a matrix
- Proofs that: $\text{Sp}(2n,\mathbb{C})$ is Lie Group and $\text{sp}(2n,\mathbb{C})$ is Lie Algebra
- Are non-standard models always not well-founded?
- How to prove that a set of logical connectives is functionally complete(incomplete)?
- Ideals in the ring of endomorphisms of a vector space of uncountably infinite dimension.
- show invariant subspace is direct sum decomposition

If you work in two-sorted first-order logic, it is possible to formalize vector spaces by using one sort for the field of scalars and another sort for the vectors, just as you describe. But there is no way, within the theory itself, to force an embedding of models to act as the identity on the scalars. For example, if you take a structure for your theory, every automorphism of the field will give a homomorphism from the structure to itself. And every proper subfield of the field of scalars gives rise to a proper substructure in which the vectors remain the same and the scalars are cut down. This is the motivation for the usual method in which the field is fixed through the addition of a lot of constant symbols to the language.

One response is to simply refuse to consider such things when you work on your proofs. That is, you could simply limit your attention to those homomorphisms that do preserve the field, and those substructures that do not change the field. It depends on what you are trying to get out of the formalization.

It’s not clear what your motivations are but, in any case, I thought it would be worthwhile to point out that usually the most convenient way to specify a vector space is to view it as a “group with operators”, i.e. for each scalar $c\:$ introduce a unary operation $\ c(v) = c\:v\:$.

- $C$ with $L^1$ norm is not Banach space.
- How big is infinity?
- What is the name of this theorem, and are there any caveats?
- What is the value of $\lim_{x\to 0}x^x$?
- Easy way to compute logarithms without a calculator?
- Infinite product equality $\prod_{n=1}^{\infty} \left(1-x^n+x^{2n}\right) = \prod_{n=1}^{\infty} \frac1{1+x^{2n-1}+x^{4n-2}}$
- How to compute the $n_{th}$ derivative of a composition: ${\left( {f \circ g} \right)^{(n)}}=?$
- Defining the Initial Conditions for a Planetary Motion to Have a Circular Orbit.
- Show $\lim_{n \to \infty} \sum_{i=1}^n Y_i/\sum_{i=1}^n Y_i^2 = 1$ for Bernoulli distributed random variables $Y_i$
- Medians of a triangle and similar triangle properties
- the real part of a holomorphic function on C \ {0, 1}
- Count the number group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$ ?
- Approximation in Sobolev Spaces
- Let $f$ be differentiable on $(0,\infty)$.Show that $\lim\limits_{x\to \infty}(f(x)+f'(x))=0$,then $\lim\limits_{x\to \infty}f(x)=0$
- Why do complex functions have a finite radius of convergence?