Suppose that $\displaystyle {v_1,v_2,......,v_n} $is a linearly independent set of vectors in $\displaystyle R^n $. Show that if A is an n x n nonsingular matrix, then $\displaystyle {Av_1,Av_2,......,Av_n} $is linearly independent.
Suppose that $\displaystyle {v_1,v_2,......,v_n} $is a linearly independent set of vectors in $\displaystyle R^n $. Show that if A is an n x n nonsingular matrix, then $\displaystyle {Av_1,Av_2,......,Av_n} $is linearly independent.
For scalars $\displaystyle a_i$, $\displaystyle 0=\sum\limits_{i=1}^na_iAv_i=\sum\limits_{i=1}^nA( a_iv_i)=A\left(\sum\limits_{i=1}^na_iv_i\right)$.
Now use that A is non-singular and then use that the given vectors are lin. indep.