Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Coordinates in a basis via the basis matrix

Builds on Coordinate vectorBasis

Statement

Let ℬ=(b1,…,bn)\mathcal B = (b_1, \dots, b_n) be an ordered basis of ℝn\R^n. Let PℬP_{\mathcal B} be the matrix whose columns are b1,…,bnb_1, \dots, b_n in this order. Then PℬP_{\mathcal B} is invertible, and for each vector x∈ℝnx \in \R^n the coordinate vector [x]ℬ[x]_{\mathcal B} satisfies x=Pℬ [x]ℬ,[x]ℬ=Pℬ−1 x.x = P_{\mathcal B}\,[x]_{\mathcal B}, \qquad [x]_{\mathcal B} = P_{\mathcal B}^{-1}\,x.

Remarks

The product Pℬ[x]ℬP_{\mathcal B}[x]_{\mathcal B} is the combination of the columns of PℬP_{\mathcal B} with the coordinates as weights. This is the expansion of xx in the basis. The second formula follows from the first by multiplying with the inverse matrix.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Basis, concept, no evidence levelBasisconceptVector 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 vectordefinitionCoordinates in a basis via the basis matrix, statement, no evidence levelCoordinates in a basi…statement
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. Basis concept
  2. Vector space definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear span definition
  7. Spanning set definition
  8. Basis definition
  9. Coordinate vector definition
  10. Coordinates in a basis via the basis matrix statement

Further reading

  1. Web https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section3-2.xml#L746, src/section3-2.xml:746
About this page
TagJ03X8QRF1F
URIm2://J03X8QRF1F
Kindstatement, theorem
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status J03X8QRF1F.
AboutCoordinate vector
UsesCoordinate vector, Basis
Changed4 November 2026

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

No page links to this page yet.