Algebra I: A Basic Course in Abstract Algebra

Chapter 16

Change of Basis

16.1 Coordinate Mapping

Let V be an n dimensional vector space and B = {v₁, v₂, …,v_n} be an ordered basis. Then, every v ∈ V can be uniquely expressed as a linear combination of elements of B, so that there exists unique elements α₁,α₂, … α_n ∈ F such that v = α₁v₁ + α₂v₂ + … α_nv_n.

Unless otherwise stated, V will denote a vector space over a field F.

This defines a mapping T : V → ℝⁿ which maps v → (α₁, α₂, …, α_n)^t It is easy to verify that T is a one-to-one linear transformation of V onto ℝⁿ. This mapping is called the coordinate mapping and we denote it by [ ]B, we write [v]B = (α₁ + α₂ + … α_n)^t.

Example16.1.   Consider V = P₃, over ℝ. Then dim V = 4, B = {1, x, x², x³} is an ordered basis for V. Let p = 2 − 3x² + ...