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:

• +: 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, ··· an) | a1, a2, ··· an ∈ F} = {$(\begin{smallmatrix}a_1\\ a_2 \\ …\\ a_n\end{smallmatrix})$| a1, a2,··· an ∈ 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 + ··· + anxn | n ≥ 0, ai ∈ F}. In particular ℤp[x] of polynomials with coefficients from ℤp, 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, ···, xn with coefficients c1, c2, ···, cn, and b1 ∈ F, F being a field, e.g., c1x1+···+cnxn=b1, d1x1+···+dnxn=b2, etc. V is a vector space. Let E1, E2, … Em be a list of linear equation with a common solution x ∈ Fn ⇒ x is obviously a solution to any linear combination of a1E1 +a2E2+···+anEn.

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. Fn[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, ···, 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.

• 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

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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