I know this sounds like a studpid question in here, but I just want to organize and clear what I studied.
For an $n\times n$ matirx $A$, it has independent columns when nullspace only has zero vector. And independent columns mean $A$ has rank $n$, therefore by the rank theorem, nullspace has zero dimension. That is, zero vector is zero dimension, is that right?
AND one more thing. I want to show that $\lbrace Av_1,...,Av_n \rbrace$ span $R^n$ when $\lbrace v_1,...,v_n \rbrace$ form a basis. Dimension theorem is used in here? If so, how can I show that $\lbrace Av_1,...,Av_n \rbrace$ span $R^n$?