Linear mapping/Matrix for basis/Surjective and column generating system/Exercise/Solution
We consider the commutative diagramm
Since the coordinate mappings are bijective, the map is surjective if and only if is surjective. The image vector of the th standard vector under is the th column of and the image space for is the space generated by the columnns. Therefore surjectivity is equivalent with the property that the columns form a generating system of .