## Chapter 16

## Change of Basis

#### 16.1 Coordinate Mapping

Let *V* be an *n* dimensional vector space and *B = {v*_{1}, v_{2}, …,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 *α*_{1},α_{2}, … α_{n} ∈ *F* such that *v = α*_{1}v_{1} + *α*_{2}v_{2} + *… α*_{n}v_{n}.

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

This defines a mapping *T* : *V →* ℝ^{n} which maps *v →* (α_{1}, α_{2}, …, α_{n})^{t} It is easy to verify that *T* is a one-to-one linear transformation of *V* onto ℝ^{n}. This mapping is called the coordinate mapping and we denote it by [ ]B, we write [v]B = (α_{1} + α_{2} + … α_{n})^{t}.

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