Vectors
2.2 Vector Spaces
Vector space axioms, linear independence, span, basis, Gram-Schmidt orthogonalization process, orthogonal and orthonormal basis.
2.2 Vector Spaces
Definition 2.9Vector Space
A set V closed under addition and scalar multiplication, satisfying eight axioms.
Addition axiomsAssociativity, null vector 0, inverse −u, and commutativity.
Scalar multiplication axiomsDistributivity, associativity, and unit scalar 1·u = u.
Example 2.5Solutions of AX = 0
Solutions X of AX = 0 form a vector space.
ProofIf AX₁ = 0 and AX₂ = 0, then A(X₁ + X₂) = 0 and A(λX₁) = 0.
Not a spaceSolutions of AX = B with B ≠ 0 do not form a vector space because A(X₁ + X₂) = 2B ≠ B.
Linear Independence
Linear combinationk₁u₁ + k₂u₂ + ... + kₘuₘ
Linearly independentk₁u₁ + ... + kₘuₘ = 0 implies k₁ = k₂ = ... = kₘ = 0.
Linearly dependentSome non-trivial combination equals zero.
Example 2.8Linear combination check
Can v = i − k be written as a linear combination of these vectors?
u₁ = i + 2j + 3k
u₂ = −i + j + k
u₃ = j + k
System1 = a − b0 = 2a + b + c−1 = 3a + b + c
Gaussian eliminationa = −1, b = −2, c = 4Yes, v is linearly representable.
Basis and Dimension
BasisA set of linearly independent vectors that span the entire space.
DimensionNumber of vectors in any basis.
Standard basis of ℝ³{i, j, k}Dimension 3.
2.3.4 Orthogonal and Orthonormal Basis
Orthogonal basisAll basis vectors are mutually perpendicular, so their dot product is zero.
Orthonormal basisOrthogonal and each vector has unit magnitude: |eᵢ| = 1
Standard basis{i, j, k} is orthonormal.
2.3.5 Gram-Schmidt Process
Converts any linearly independent set {v₁, v₂, ...} into an orthonormal basis {e₁, e₂, ...}.
1
u₁ = v₁e₁ = u₁|u₁|
2
u₂ = v₂ − (v₂·e₁)e₁e₂ = u₂|u₂|
3
u₃ = v₃ − (v₃·e₁)e₁ − (v₃·e₂)e₂e₃ = u₃|u₃|
At each step: subtract projections onto already-found orthonormal vectors, then normalize. The result is a set of mutually perpendicular unit vectors spanning the same space.