I know how to use methods like system of linear equations to show that any vector in R^n can b

Theresa Chung 2022-02-24 Answered
I know how to use methods like system of linear equations to show that any vector in Rn can be expressed in the form of unique linear combinations of a given set of n linearly independent vectors, but how to expand this result to that any given set of exactly n linearly independent vectors must form a basis of Rn space? How to understand it conceptually?
I saw there are similar questions, but the answers in those posts do not really convince me.
Thanks in advance.
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Expert Answer

mtakadamu9i5
Answered 2022-02-25 Author has 8 answers
I'm not sure if this answers your question but in order for a set of vectors to be a basis of a vector space it has to be linear independant and has to span the entire space. Now if you're given a set of n linearly independant vectors in Rn you only have to show that it spans Rn. But as you said yourself this is indeed the case since you can express any vector as a linear combination of vectors from your set.
If that isn't satisfactory, you could also think about it this way: Suppose you have n linearly independant vectors (v1,,vn) and another vector w not in the span of your (v1,,vn). Then (v1,,vnw) is linear dependant since it's a set of n+1 vectors in a vector space of dimension n. This means there are μ,λiR,i=1,,n not all 0 with 0=μw+i=1nλivi. Now μ0 because otherwhise (v1,,vn) would be linear dependant. Therefore w=1μi=1nλivi but this means w is in the span of (v1,,vn) a contradiction.
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