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## Change of Basis

#### 16.1 Coordinate Mapping

Let V be an n dimensional vector space and B = {v1, v2, …,vn} 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 α12, … αnF such that v = α1v1 + α2v2 + … αnvn.

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

This defines a mapping T : V →n which maps v →1, α2, …, αn)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 + … αn)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 + ...

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