Let S = {v_1, v_2, ..., v_n} be a finite set of vectors in a vector space V. Show that S is a basis for V iff every member of V can be written uniquely as a linear combination of the vectors in S.

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- November 29th 2009, 11:25 PMCoda202Vector Spaces and Basis
Let S = {v_1, v_2, ..., v_n} be a finite set of vectors in a vector space V. Show that S is a basis for V iff every member of V can be written uniquely as a linear combination of the vectors in S.

- November 29th 2009, 11:31 PMaman_cc
- November 30th 2009, 05:39 AMHallsofIvy
In particular, to prove the vectors are independent, saying " every member of V can be written uniquely as a linear combination of the vectors in S" means that the

**zero**vector can be written**uniquely**as such a linear combination. There is one "obvious" linear combination and now you know it is the only one. - November 30th 2009, 10:34 AMRaoh
suppose is a basis of V.

in case

is the given basis.

let's take any vector of

or

We have,

+ +

gives , and since the system is independent .

Therefore, and .