Linear Maps 7 (Ordered Basis and System of Coordinates)

In this episode, we introduce the notion of an ordered basis with examples. Then we show how an ordered basis of V sets up an isomorphism with R^n where n= dim V. We then show how to relate the systems of coordinate associated with different ordered bases. To establish the relation, we define the Transition matrices. We illustrate all these by means of well-chosen simple examples.

It is worth learning this very well. In view of machine learning etc., the computational aspects of linear algebra become very crucial. The system of coordinates is the medium when abstract vector spaces are `identified' with some R^n so that we can do computations.

Timestamp provided by Ihwarya.
00:00 Introduction
2:08 Discussion about ordered basis
6:14 Coordinate map associate to ordered basis
9:25 Any Finite dimensional vector space is linearly isomorphic to Rn
15:21 Coordinate map for two different ordered bases of V
16:44 Examples
28:51 Relation between the coordinate map of two different bases in V
40:18 Transition matrix from B to B'
45:22 Some Examples
50:47 Brainstorming activity

[Durgaprasad_Nayak]

Sir in 26:24 phi B' (x²) = (a², 2a, 1)