If we have a set of linearly independent vectors, can they be mapped to a set of linearly dependent vectors? and vice versa. for example,
Given T : V → W is a linear map where $T(v_1) = w_1,··· ,T(v_n) = w_n$ for some vectors $w_1,··· ,w_n ∈ W$.
I think if the vectors {$w_1,...,w_n$} are linearly independent then it doesn't necessarily mean that the vectors {$v_1,...,v_n$} must also be linearly independent. If we let $v_1= c_1v_2+...+c_{n-1}v_n$ , for some scalars $c$, and similarly write the other vectors in $V$ as linear combinations of each other, then {$v_1,...,v_n$} would be dependent but would still be sent to linearly independent vectors in $W$.
Am I on the right track or is there a flaw in my logic/math. Thanks!