Editing Quotient space (linear algebra) (section)

===Subspaces of Cartesian Space===

Another example is the quotient of '''R'''<sup>''n''</sup> by the subspace spanned by the first ''m'' [[standard basis vector]]s.  The space '''R'''<sup>''n''</sup> consists of all ''n''-tuples of [[real number]]s {{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}}.  The subspace, identified with '''R'''<sup>''m''</sup>, consists of all ''n''-tuples such that the last ''n'' − ''m'' entries are zero: {{nowrap|(''x''<sub>1</sub>, ..., ''x''<sub>''m''</sub>, 0, 0, ..., 0)}}.  Two vectors of '''R'''<sup>''n''</sup> are in the same equivalence class modulo the subspace [[if and only if]] they are identical in the last ''n'' − ''m'' coordinates. The quotient space  '''R'''<sup>''n''</sup>/'''R'''<sup>''m''</sup> is [[isomorphic]] to  '''R'''<sup>''n''−''m''</sup> in an obvious manner.