Question 35787: Let V be a linear subspace of R^n. Suppose m vectors v1,...,vm span V. Prove that there is a subset of {v1,...,vm} which form a basis for V.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Let V be a linear subspace of R^n. Suppose m vectors v1,...,vm span V.
THAT IS EVERY VECTOR IN V COULD BE WRITTEN AS A LINEAR COMBINATION OF V1,V2,...VM
NOW THERE ARE ARE 2 POSSIBILITIES.
1.V1,V2....VM ARE INDEPENDENT...THEN SINCE THEY SPAN V AND ARE INDEPENDENT THEY A SUBSET OF V1,V2,....VM FORM A BASIS FOR V BY DEFINITION.
2.V1,V2,...VM ARE DEPENDENT SET.
THEN WE CAN WRITE ONE OF THEM AS A LINEAR COMBINATION OF OTHERS.WE CAN THEN REMOVE THAT DEPENDENT VECTOR SAY VP.
IF THE REST ARE STILL DEPENDENT ,WE CAN REPEAT THE PROCESS TILL WE REMOVE ALL THE DEPENDENT VECTORS AND REMAIN WITH AN INDEPENDENT SUBSET OF V1,V2,.....VM...WHICH COULD STILL EXPRESS EACH AND EVERY VECTOR IN V BY A LINEAR COMBINATION OF THOSE VECTORS FORMING THE SUBSET.HENCE THEY BEING INDEPENDENT FORM A BASIS FOR V.
Prove that there is a subset of {v1,...,vm} which form a basis for V.
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