linear algebra - Does a set of basis vectors have to be linearly independent?

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