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Linear Algebra, Optimisation and Numerical Methods / Linear Transformations, Matrix Representation, and Rank-Nullity

The Matrix of a Linear Transformation and Change of Basis

The previous note built the standard matrix of a linear map from its action on the standard basis. But the standard basis is only one choice — and often not the best one. This note shows how the same transformation looks different when described in a different basis, gives the formula for converting between representations, and introduces similarity, A′=P−1APA' = P^{-1}AP, the relation between any two matrices representing the same underlying map. Similarity is the concept that makes diagonalisation (Stage 3) a well-posed question: "does there exist a basis in which this transformation's matrix is diagonal?"
Learning objectives. By the end of this note you should be able to:
  • convert a vector's coordinates between two different bases;
  • construct the matrix of a linear map relative to a chosen (non-standard) basis;
  • state and use the similarity relation A′=P−1APA'=P^{-1}AP;
  • explain, informally, why a well-chosen basis can make a transformation's matrix far simpler.

In this lesson

  1. Coordinates relative to a basis
  2. The matrix of TT relative to a chosen basis

Before you start

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