Math 2.0

Linear and multilinear algebra; matrix theory Definition

Change of basis matrix

Builds on BasisCoordinate vector

Definition

Let FF be a field and let VV be a vector space over FF with two bases B=(b1,…,bn)B = (b_1, \dots, b_n) and B′=(b1′,…,bn′)B' = (b'_1, \dots, b'_n). The change of basis matrix from BB to B′B' is the n×nn \times n matrix P=(pij)P = (p_{ij}) defined by bj′=∑i=1npijbi(1≤j≤n).b'_j = \sum_{i=1}^{n} p_{ij} b_i \qquad (1 \le j \le n). Its column jj is the coordinate vector [bj′]B[b'_j]_B.

Remarks

For each v∈Vv \in V, [v]B=P[v]B′[v]_B = P [v]_{B'}. The matrix PP is invertible, and P−1P^{-1} is the change of basis matrix from B′B' to BB. PP is the matrix of the identity map of VV in the bases B′B' (domain) and BB (codomain). Conventions differ between authors; some call P−1P^{-1} the change of basis matrix.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionChange of basis matrix, definition, no evidence levelChange of basis matrixdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Vector space definition
  2. Linear combination definition
  3. Linear subspace definition
  4. Linear independence definition
  5. Linear span definition
  6. Spanning set definition
  7. Basis definition
  8. Coordinate vector definition
  9. Change of basis matrix definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Change_of_basis, Change-of-basis formula
About this page
Tag0KWXYZPJKN
URIm2://0KWXYZPJKN
Kinddefinition
WritingW0 stub
MSC15A03, 15A04
DefinesChange of basis matrix
UsesBasis, Coordinate vector

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by