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MATH
[1251]

Corollaries from Coordinates and bases

All corollaries from Coordinates and bases.

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Square-matrix basis criteria

For a square matrix AA with columns u1,…,unu_1,\ldots,u_n, the following are equivalent:

  • the columns are linearly independent;
  • the columns span Rn\mathbb R^n;
  • the columns form a basis for Rn\mathbb R^n.

For a square matrix, these basis criteria are equivalent to the familiar tests for invertibility.

  1. [1]:Basis
  2. [2]:Counting with dimension
  3. [3]:The homogeneous-system test

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Square-matrix basis criteria