The definition for a set of vectors to be considered a basis for $R^n$ is that 1) this set spans $R^n$ - any vector in $R^n$ can be written as a combination of this set and 2) this set is linearly independent.
Extending this analogy to vector spaces, $V$, from the below article, it states "a set B of elements (vectors) in a vector space V is called a $\textbf{basis}$, if every element of V may be written in a unique way as a (finite) linear combination of elements of B."
Then, it goes on to say "B is a $\textbf{basis}$ if its elements are linearly independent and every element of V is a linear combination of elements of B. In more general terms, a basis is a linearly independent spanning set."
https://en.wikipedia.org/wiki/Basis_(linear_algebra)
So, is a set $B$ considered a basis if everything in $V$ can be written with $B$ and $B$ must be linearly independent? Or, is $B$ a basis solely based on the fact that everything in $V$ can be written with $B$?
Answer
To remove any confusion:
A Basis $B$ of a vector space $V$ is a subset $B \subset V$ such that
- $span(B) = V$ and
- $B$ is linearly independent
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