I know this sounds like a stupid question, but I just want to organize and clear what I studied.
For an $n\times n$ matrix $A$, 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 $R^n$?
Best Answer
Yes but here's a minor nit pick: A vector doesn't have a dimension, you want to say that the subspace spanned by the zero vector has dimension zero.
For the second question you may use the rank nullity theorem. You have $\mathrm{dim} \mathrm{ker} A = 0$ and hence $\mathrm{dim} \mathrm{im} A = \mathrm{dim}R^n - \mathrm{dim} \mathrm{ker} A = \mathrm{dim}R^n$ hence the $v_i$ span $R^n$.