Editing Linear subspace (section)

===Coordinates for a vector===
:'''Input''' A basis {'''b'''<sub>1</sub>, '''b'''<sub>2</sub>, ..., '''b'''<sub>''k''</sub>} for a subspace ''S'' of ''K''<sup>''n''</sup>, and a vector {{nowrap| '''v''' ∈ ''S''}}
:'''Output''' Numbers ''t''<sub>1</sub>, ''t''<sub>2</sub>, ..., ''t''<sub>''k''</sub> such that {{nowrap|1= '''v''' = ''t''<sub>1</sub>'''b'''<sub>1</sub> + ··· + ''t''<sub>''k''</sub>'''b'''<sub>''k''</sub>}}
:# Create an [[augmented matrix]] ''A'' whose columns are '''b'''<sub>1</sub>,...,'''b'''<sub>''k''</sub> , with the last column being '''v'''.
:# Use elementary row operations to put ''A'' into reduced row echelon form.
:# Express the final column of the reduced echelon form as a linear combination of the first ''k'' columns. The coefficients used are the desired numbers {{nowrap| ''t''<sub>1</sub>, ''t''<sub>2</sub>, ..., ''t''<sub>''k''</sub>}}. (These should be precisely the first ''k'' entries in the final column of the reduced echelon form.)
If the final column of the reduced row echelon form contains a pivot, then the input vector '''v''' does not lie in ''S''.