Definition of ℝ_n

The set of all n-tuples of real numbers is denoted by ℝ_n:

u = (a1, a2, ..., an)

• Each u is called a point or vector.
• The components a_i are called coordinates, components, entries, or elements.
• The term scalar is used for elements of ℝ.

Equality of Vectors and the Zero Vector

• Two vectors u and v are equal (u = v) if:
  • They have the same number of components, and
  • Corresponding components are equal.
• Example: (1,2,3) ≠ (2,3,1) even though they contain the same numbers.
• The vector (0, 0, ..., 0) is called the zero vector, denoted by 0.

Example of Vectors

The following are vectors:

• (2, -5), (7, 9) → elements of ℝ₂
• (0, 0, 0) → zero vector in ℝ₃
• (3, 4, 5) → vector in ℝ₃
• The first two belong to ℝ₂, the last two to ℝ₃.

Definition of a Vector Space

Let V be a nonempty set with two operations:

• Vector Addition: Assigns to any u, v ∈ V a sum u + v ∈ V.
• Scalar Multiplication: Assigns to any u ∈ V, k ∈ K a product ku ∈ V.

Then V is a vector space over the field K if the following axioms hold for all vectors u, v, w ∈ V:

Vector Space Axioms

• [A1] (u + v) + w = u + (v + w)
• [A2] There exists 0 ∈ V such that u + 0 = 0 + u = u
• [A3] For each u ∈ V, there exists -u ∈ V such that u + (-u) = (-u) + u = 0
• [A4] u + v = v + u
• [M1] k(u + v) = ku + kv, for k ∈ K
• [M2] (a + b)u = au + bu, for a, b ∈ K
• [M3] a(bu) = (ab)u, for a, b ∈ K
• [M4] 1u = u, for the unit scalar 1 ∈ K

Remarks

• The additive structure forms a commutative group.
• Vector sum v₁ + v₂ + … + v_n is associative.
• Zero vector is unique, and each vector has a unique negative.
• If u + w = v + w, then u = v (Cancellation Law).
• Subtraction: u – v = u + (-v)

Vector Space: Kⁿ

Let K be a field. Then Kⁿ is the set of all n-tuples of elements in K:

Kn = {(a1, a2, ..., an) | ai ∈ K}

• Vector Addition: (a1, ..., an) + (b1, ..., bn) = (a1 + b1, ..., an + bn)
• Scalar Multiplication: c(a1, ..., an) = (ca1, ..., can)
• Zero vector: (0, 0, ..., 0)
• Negative of a vector: -(a1, ..., an) = (-a1, ..., -an)

Polynomial Space P(ℝ)

Let P(ℝ) be the set of all polynomials:

p(x) = a0 + a1x + a2x2 + ... + anxn

• Vector Addition: (p + q)(x) = p(x) + q(x)
• Scalar Multiplication: (ap)(x) = a ċ p(x)
• Zero polynomial is 0(x) = 0

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