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 $B$?
Best Answer
The section where it says "written in a unique way" is equivalent to linear independence. I think this is where some confusion may have arisen.
Your first concluding statement is correct. The second isn't. The basis must be linearly independent; either said explicitely or with the magic word "uniquely" added to your second statement to make it implicitely so.