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, ℤ_p (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:

• +: VxV → V, ∀v, w ∈ V, v + w ∈ V
• ·: FxV → V, ∀v ∈ V, a ∈ F, av ∈ V. This operation is called scalar multiplication

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:

• In physics, a vector represents a quantity that has both magnitude and direction. Vectors can be added together by the parallelogram rule. Place both vectors so they have the same initial point, then draw a parallel line across from the first vector. Next, draw a parallel line across from the second vector and you will form a parallelogram. Finally, draw a straight line from the tails of the two vectors across the diagonal of the parallelogram.

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

• The trivial vector space, V = {0} over any field. The scalar multiplication is defined as a0 = 0 ∀a ∈ F.
• Let F be ℚ, ℝ, ℂ, etc. and V = F. In other words, a field F is a vector space over itself where vector addiction and the scalar multiplication is the addition and multiplication in the field respectively.
• Let F be any field, V = F x F = {(a₁, a₂) | a₁, a₂ ∈ F} = {$(\begin{smallmatrix}a_1\\ a_2\end{smallmatrix})$| a₁, a₂ ∈ F} is a vector space where v + w = $(\begin{smallmatrix}a_1\\ a_2\end{smallmatrix}) + (\begin{smallmatrix}b_1\\ b_2\end{smallmatrix}) = (\begin{smallmatrix}a_1+b_1\\ a_2+b_2\end{smallmatrix})$ and αv= $α(\begin{smallmatrix}a_1\\ a_2\end{smallmatrix}) = (\begin{smallmatrix}αa_1\\ αa_2\end{smallmatrix})$ We can generalize it, with V = Fⁿ {(a₁, a₂, ··· a_n) | a₁, a₂, ··· a_n ∈ F} = {$(\begin{smallmatrix}a_1\\ a_2 \\ …\\ a_n\end{smallmatrix})$| a₁, a₂,··· a_n ∈ F}. In particular, ℝⁿ is a vector space over ℝ.
• Let F be any field, the vector space of polynomials with coefficients from F, F[x]={a₀ + a₁x + ··· + a_nxⁿ | n ≥ 0, a_i ∈ F}. In particular ℤ_p[x] of polynomials with coefficients from ℤ_p, p prime, is a vector space.
• Let F = ℝ and F(ℝ) be the set of al real-valued functions from ℝ → ℝ. It is a vector space over ℝ where (f+g)(x) = f(x)+g(x) and (αf)(x) = αf(x).
• The set of complex numbers ℂ is a vector space over ℝ.
• Let V be the set of linear equations with n variables, say x₁, x₂, ···, x_n with coefficients c₁, c₂, ···, c_n, and b₁ ∈ F, F being a field, e.g., c₁x₁+···+c_nx_n=b₁, d₁x₁+···+d_nx_n=b₂, etc. V is a vector space. Let E₁, E₂, … E_m be a list of linear equation with a common solution x ∈ Fⁿ ⇒ x is obviously a solution to any linear combination of a₁E₁ +a₂E₂+···+a_nE_n.

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

• The zero vector 0 is unique. The additive inverse of u is unique.

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

• α·0 = 0. α.0 = α·(0 + 0) = α·0 + α·0 ⇒ [Cancellation law] 0 = α·0.
• 0·v = 0. It is quite similar.
• (-α)v = -(αv). Proof: αv + (-α)v = (α + (-α))v = 0v = 0 ⇒ αv + (-α)v = 0 ⇒ [-(αv) in both sides] (-α)v = -(αv)
• αv = 0 ↭ α = 0 or v = 0. Proof: ⇐) Trivial. ⇒. Suppose αv = 0. If α = 0, we are done. Suppose that α ≠ 0, α ∈ F ⇒ [F field, every non-zero element has a multiplicative inverse] ∃ α^-1 ∈ F: αα^-1 = 1 ∈ F. αv = 0 ⇒ α^-1(αv) = α^-10 = 0 ⇒ v = 0.

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,

• (U, +) is a subgroup of (V, +), that is, 0 ∈ U; ∀u, v ∈ U, u + v ∈ U and -u ∈ U.
• ∀u ∈ U, ∀α ∈ F, αu ∈ U.

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

Examples.

• Let F be any field, V = F x F = {(a₁, a₂) | a₁, a₂ ∈ F} = {$(\begin{smallmatrix}a_1\\ a_2\end{smallmatrix})$| a₁, a₂ ∈ F}, U = {$(\begin{smallmatrix}a\\ 0\end{smallmatrix})$| a ∈ F} is a subspace of V.
• Let F be any field, V = F x F = {(x₁, x₂) | x₁, x₂ ∈ F} = {$(\begin{smallmatrix}x_1\\ x_2\end{smallmatrix})$| x₁, x₂ ∈ F}, U = {$(\begin{smallmatrix}x_1\\ x_2\end{smallmatrix})$| x₁, x₂ ∈ F, ax₁ + bx₂ = 0, where a and be are elements that are fixed from the field, a, b ∈ F} is a subspace of V. In particular, V = ℝ², U = {(x₁, x₂) | 2x₁ - x₂ = 0}
• The subset of F(ℝ) that contains the set of real-valued continuous functions, C⁰(ℝ) is a vector subspace of F(ℝ). Let Cⁿ be the set of real-valued continuous and differentiable functions that are n-times differentiable. The set of real-valued continuous and differentiable functions is a subspace of the vector space of real-valued continuous functions. Futhermore, C⁰(ℝ) ⊇ C¹(ℝ) ⊇ C²(ℝ) ⊇ ··· this a string of vector subspaces.
• Let F[x] be the vector space of all coefficients in F. F_n[x] be the set of all polynomials of degrees less or equal to n is a subspace of F[x]. In particular, the sets {a₁x + a₀ | a₁, a₀ ∈ ℝ} and {a₂x² + a₁x + a₀ | a₂, a₁, a₀ ∈ ℝ} are subspaces of ℝ[x].

  We need to ask for all polynomials of degree less than n and not just of degree equal n, because it needs to be close under addition, e.g., (x² + 2) + (-x² -7x +4) = -7x +6. Futhermore, we need to include the zero polynomial.

• W, W’ ⊆ C¹(ℝ) -real functions continuous and diferenciables-, W = {f | f(3) = 0}, W’ = {f | f’(x) = f(x)}, W and W’ are subspaces of C¹(ℝ).

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

Proof: v = α₁v₁ + α₂v₂ + ··· + α_nv_n, w = β₁v₁ + β₂v₂ + ··· + β_nv_n

v + w = (α₁ + β₁)v₁ + (α₂ + β₂)v₂ + ··· + (α_n + β_n)v_n ∈ ⟨v₁, v₂, ···, v_n⟩ because α_i + β_i ∈ F, F field.

αv = (αα₁)v₁ + (αα₂)v₂ + ··· + (αα_n)v_n ∈ ⟨v₁, v₂, ···, v_n⟩ because αα_i ∈ F, F field.

Exercises.

• In ℂ², 3 $(\begin{smallmatrix}2\\ i\end{smallmatrix})+2i(\begin{smallmatrix}2+i\\ 1+i\end{smallmatrix})=(\begin{smallmatrix}6+2i(2+i)\\ 3i+2i(1+i)\end{smallmatrix})=(\begin{smallmatrix}4(1 + i)\\ -2 + 5 i\end{smallmatrix})$

• In $ℤ_3^3,~ (\begin{smallmatrix}0\\ 2\\ 1\end{smallmatrix})+2(\begin{smallmatrix}2\\ 1\\ 1\end{smallmatrix})+2(\begin{smallmatrix}1\\ 0\\ 1\end{smallmatrix})=(\begin{smallmatrix}0+1+2\\ 2+2\\ 1+2+2\end{smallmatrix})=(\begin{smallmatrix}0\\ 1\\ 2\end{smallmatrix})$

Bibliography

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.