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, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, Andrew Misseldine, blackpenredpen, and MathMajor, YouTube’s channels.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. MIT OpenCourseWare, 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007, YouTube.
8. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.