JustToThePoint English Website Version
JustToThePoint en español
JustToThePoint in Thai

Vector Spaces

The more I learn about people, the more I like dogs and I don’t even have a dog, Apocalypse, Anawim, #justtothepoint.

Recall that a field is a commutative ring with unity such that each nonzero element has a multiplicative inverse, e.g., every finite integral domain, ℤp (p prime), ℚ, ℝ, and ℂ. A field has characteristic zero or characteristic p with p prime and F[x] is a field.

 

Definition. Let F be a field and V a set of two operations:

We say V is a vector space over F if (V, +) is an Abelian group under addition, and ∀a, b ∈ F, u, v ∈V the following conditions hold:

  1. a(v + u) = av + au, (a + b)v = av + bv
  2. a(bv) = (ab)v
  3. 1v = v.

Examples:

Let $u = (\begin{smallmatrix}2\\ 5\\ 4\end{smallmatrix}), v = (\begin{smallmatrix}1\\ 3\\ 2\end{smallmatrix})∈ ℝ^3.~ u+v=(\begin{smallmatrix}2\\ 5\\ 4\end{smallmatrix})+(\begin{smallmatrix}1\\ 3\\ 2\end{smallmatrix})=(\begin{smallmatrix}3\\ 8\\ 6\end{smallmatrix}).~ 4u=4(\begin{smallmatrix}2\\ 5\\ 4\end{smallmatrix})=(\begin{smallmatrix}8\\ 20\\ 16\end{smallmatrix})$

 

Proposition. Suppose V is a a vector space, K a field, α ∈ F, v ∈ V. Then,

Suppose there is another zero, say θ, ∀v ∈ V: θ + v = v + θ = v. In particular θ + 0 = 0 = [∀v∈ V: 0 + v = v + 0 = v. In particular, 0 + θ = θ + 0 = θ.] θ

Similarly, ∃u’ and -u inverses: u’ = u’ + 0 = [-u is an inverse of u] u’ + (u -u) = [Associative] (u’ +u) -u = [u’ is “another” inverse of u] 0 -u = -u

Definition. Suppose V is a vector space over a field F and let U be a subset of V (U ⊆ V). We say U is a subspace of V if U is also a vector space over F under the operations of V or equivalently if,

Proposition. U ⊆ V is a subspace iff ∀α ∈ F, u, v ∈ U, we have αu, u + v ∈ U.

Examples.

Definition. Suppose V is a vector space over F, α1, α2, ···, αn ∈ F, v1, v2, ···, vn ∈ V, then the linear combination of v1, v2, ···, vn with weights or coefficients α1, α2, ···, αn is α1v1 + α2v2 + ··· + αnvn. The set of all such lineal combinations is called the subspace of V spanned by v1, v2, ···, vn, that is ⟨v1, v2, ···, vn⟩ = span{v1, v2, ···, vn} = {α1v1 + α2v2 + ··· + αnvn| αi ∈ F}

Proof: v = α1v1 + α2v2 + ··· + αnvn, w = β1v1 + β2v2 + ··· + βnvn

v + w = (α1 + β1)v1 + (α2 + β2)v2 + ··· + (αn + βn)vn ∈ ⟨v1, v2, ···, vn⟩ because αi + βi ∈ F, F field.

αv = (αα1)v1 + (αα2)v2 + ··· + (ααn)vn ∈ ⟨v1, v2, ···, vn⟩ because ααi ∈ F, F field.

Exercises.

Bibliography

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. This post relies heavily on the following resources, specially on NPTEL-NOC IITM, Introduction to Galois Theory, Michael Penn, and Contemporary Abstract Algebra, Joseph, A. Gallian.
  1. NPTEL-NOC IITM, Introduction to Galois Theory.
  2. Algebra, Second Edition, by Michael Artin.
  3. LibreTexts, Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
  4. Field and Galois Theory, by Patrick Morandi. Springer.
  5. Michael Penn (Abstract Algebra), and MathMajor.
  6. Contemporary Abstract Algebra, Joseph, A. Gallian.
  7. Andrew Misseldine: College Algebra and Abstract Algebra.
Bitcoin donation

JustToThePoint Copyright © 2011 - 2024 Anawim. ALL RIGHTS RESERVED. Bilingual e-books, articles, and videos to help your child and your entire family succeed, develop a healthy lifestyle, and have a lot of fun.

This website uses cookies to improve your navigation experience.
By continuing, you are consenting to our use of cookies, in accordance with our Cookies Policy and Website Terms and Conditions of use.