What is the rank in each of the following case? The columns of $A$ are $n$ vectors from $R^m$. If they are linearly independent, what is the rank of $A$? If they span $R^m$, what is the rank? If they are a basis for $R^m$ what then?
Here's my explanation:
The $n$ vectors are linearly independent, so $rank(A)=n$.
Now comes my confusion, to span $R^m$, I need $m$ vectors, so $rank(A)=m=n$. 
Same condition to be a  basis of $R^m$. Am I correct?

Yes, you are correct. Note that if the columns of $A$ are $n$ linerly independent vectors of $\mathbb{R}^m$ than $n\le m$.
