Hi: I'm going through Berberian's text and trying to do some of the exercises. For this one, I looked through old threads and the internet but I was not able to find any help. In fact, although I like the book up to this point, I can't even find a method or framework for approaching the question. Usually, the author provides a lot of examples but not for this particular exercise. So, I have tried but I don't have anything to physically show. I do understand what injective, surjective and bijective mean. The exercise is the following.
Let $T: V \rightarrow W$ be a linear mapping. Let $S$ denote a system of generators for $V$ ( I finally understand what they are based on some old threads on this list ) and let $F$ be an independent subset of $V$. Also, let $[S]$ denote the subspace generated by S. Then:
(i) If $T$ is surjective, $T(S)$ is a system of generators for $W$.
(ii) If $T$ is injective, $T(F)$ is an independent subset of $W$.
(iii) If $S$ is an arbitary subset of $V$, $[T(S)] = T([S])$.
Thanks for your help in advance.
NOTE: The book defines dependence (and therefore independence) in the following way:
Let $S$ be a set of vectors in a space V. A vector $x$ is said to be linearly dependent on S when there exist vectors $x_{1}, \ldots x_{n}$ in $S$ such that $x$ is a linear combination of the $x_{k}$. Briefly: $x$ depends on $S$.
For i), if $T$ is surjective, then each $w \in W$ can be expressed as $T(v)$ for some $v \in V$. Then, write $v = \sum_i c_ix_i$ where $x_i$ are the generators of $v$. Since $T$ is linear, we have $T(v) = \sum_ic_iT(v_i) = w$. So, the set of $T(v_i)$ is a system of generators for $W$ (any $w \in W$ can be expressed as a linear combination of the $T(v_i)$). Note that since $T$ need not be injective, this generating set may be bigger than needed.
For ii), if $T$ is injective, then $T(v_1) = T(v_2)$ implies that $v_1 = v_2$. Let $F = \{x_1, \dots, x_n\}$ be an independent set in $V$. Then, suppose for the sake of contradiction that $T(F)$ is dependent. So, WLOG we can express $T(v_1) = \sum_{i \neq 1} c_i T(v_i) = T(\sum_{i \neq 1} c_i v_i)$ for some coefficients $c_i$ not all zero. Then, by the definition of injective, $v_1 = \sum_{i \neq 1} c_i v_i$ which implies that $F$ was not an independent set, a contradiction.
For iii), let $S = \{x_1, \dots, x_n\}$ be an arbitrary subset of $V$. Let $x \in [T(S)]$, so $x = \sum_i c_i T(x_i) = T(\sum_i c_i x_i) \in T([S])$. Hence, $[T(S)] \subset T([S])$. Conversely, let $x \in T([S])$ so that $x = T(\sum_i c_i x_i) = \sum_i c_i T(x_i) \in [T(S)]$. Hence, $T([S]) \subset [T(S)]$. Thus, the two are equal.