Transformations and Dependence 
Hi, for these problems I generally get the gist of it. If you have some linearly dependent vectors $v_1, \ldots, v_m$ in $\mathbb{R}^n$ then when you transform those vectors $T(v_1), \dots, T(v_m)$ they will still be dependent, no matter what space they are in. 
And for the second problem, if $n < p$ and $m < p$ then the $T(v_1), \ldots, T(v_m)$ will have $p-m$ more vectors than a basis in space $P$ and as such may not necessarily be independent. 
First, is that correct logic? Second, this might be me overthinking, but what about the scenario where we have $v_1, \ldots, v_n vectors$ that are independent in $\mathbb{R}^n$, and we then then move to $\mathbb{R}^p$. Where $p < n$, $T(v_1), \ldots, T(v_n)$ are dependent by the logic of the answer to 37. Then, if we move back to $\mathbb{R}^n$ wouldn't that imply that $v_1, \ldots, v_n$ are dependent by the logic of the answer to 36? I am clearly missing something, any insight would be great.
Thanks
 A: Your answer to question $36$ is correct.
For the question $37$, your justification is neither clear nor correct: "if n < p and m < p then the T(v1),...,T(vm) will have p-m more vectors then a basis in space P and as such may not necessarily be independent". You're saying that if you have a family of vectors in $\mathbb{R}^p$ which cardinality is less than $p$ then it might not be independent, i.e, it can be dependent as it can be independent. It seems to me you just said a tautology: if $P$ is a proposition then we have $Q$ or $\lnot{Q}$. It's like someone saying "if the cardinality of a family of vectors in a finite dimensional space is greater than $\dim{E}$ then this family may not be independent". This statement gave us no information (in fact this family MUST be dependent). So in your case you have to PROVE that we can have both by giving examples. Otherwise, without comparing the numbers $n$, $m$ and $p$, you can actually prove easly that what's in $37$ is incorrect: simply take easy examples like the null map.
