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Vector Spaces

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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.


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.


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.



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.
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