Since I am not good at English, I am confused of what the following sentence means from T/F question.
If T is linear map, then T carries linearly independent subsets of V onto linearly independent subsets of W
I don't understand what carry onto means exactly. Does it mean that if T has independent subsets of V and there is a linearly independent subset of W? You don't have to give me the answer whether it is true or false because it is a homework.
Thank you.
I will rephrase the question in the usual symbolic notation of linear algebra.
"True or False: Let $V,W$ be vector spaces. If $T: V \to W$ is a linear transformation, then for any linearly independent subset $\{\vec{v}_1, \dots, \vec{v}_n\}$ in $V$, the set $\{T(\vec{v}_1), \dots , T(\vec{v}_{n}) \}$ is linearly independent in $W$."
Note: I am merely rephrasing what the true or false question is asking. I am not asserting that it is true or false. That is for you to decide :)
Other Note: The word "onto" is typically used in math (and particularly often in intro linear algebra courses in the US) to mean "surjective", but I don't think the author of this question intended that. If "onto" was intentionally used, and intended to mean surjective, I would interpret the question as asking if the linear transformation $T$ was a surjective function when restricted to the set of all linearly independent subsets of $V$, as a map to the set of all linearly independent subsets of $W$. This interpretation still makes mathematical sense as a true or false question, but I would be surprised if this was the intended interpretation.