Let $W$ be a vector space over $R$ and let $T:R^{6}\to W$ be a linear transformation,such that $S={[Te_2,Te_4,Te_6]}$ spans $W$.Which one of the following must be true?
$1.S$ is a basis of $W$.
$2.$$T(R^{6})\ne W$
$3.${${Te_1,Te_3,Te_5}$}spans $W$.
$4.$ker$(T)$ contains more than one element.
I am sure second option isn't correct if $T(a,b,c,d,e,f)=a$ but can't think about others.
Best Answer
It is clear that 4. holds, by the rank-nullity theorem. All others are false.