Linear maps, vectors, injectivity and surjectivity question Let $\phi : A \rightarrow B$ be a linear map and $a_{1},...,a_{n} \in A$
i) If $a_{1},...,a_{n}$ span $A$ and $\phi$ is surjective, show that $\phi(a_{1}),...\phi(a_{n})$ span $B$.  Is this also true when $\phi$ is not surjective?
ii) If $a_{1},...,a_{n}$ are linearly independent and $\phi$ is injective, show that $\phi(a_{1}),...\phi(a_{n})$ are linearly independent. Is this also true when $\phi$ is not injective?
I don't know where to begin. Any help would be appreciated.
 A: Hint for 1) For a generic element $b\in B$, find an element of $A$ in terms of the generating set that maps to $b$. Then think about the linearity of $\phi$.
Hint for 2) Look at a linear combination of the $\phi(a_i)$ equal to zero and apply linearity of $\phi$. (It is especially helpful for this half to know that a linear map is injective iff $\phi(x)=0$ implies $x=0$.)
A: for a  $<a_1,a_2,...,a_n>=A $   we want show that   $$B=<\phi (a_{1}),...,\phi(a_{n})>$$ clearly $$<\phi (a_{1}),...,\phi(a_{n})>\subset  B$$ now prove$$ B\subset <\phi (a_{1}),...,\phi(a_{n})> $$ let $y\in B$ $ \exists x\in A$ such that $\phi(x) =y$ where  $$x\in<a_1,a_2,...,a_n>  ,\exists c_i \ s.t \quad x=\sum_{i=o}^nc_ix_i$$  hence $$\phi(x) =\sum_{i=o}^nc_i\phi(x_i)$$
$$y\in<\phi (a_{1}),...,\phi(a_{n})>$$
for b $c_1\phi (a_{1})+...+c_n\phi(a_{n})$=0then $$\phi (c_1a_{1}+...+c_na_{n})=0$$  therefore $$c_1a_{1}+...+c_na_{n}=0$$ 
A: 1) So you want to show that $\phi(a_i)$ span $B$. Let $b\in B$. Then there is a
$$
a = c_1a_1 + \dots + c_na_n \in A
$$
such that $\phi(a) = b$ (because $phi$ is surjective. But that means that 
$$
b = c_1\phi(a_1) + \dots + c_n\phi(b_n).
$$
2) Say that 
$$
c_1\phi(a_1) + \dots+c_n\phi(a_n) = 0.
$$
Then
$$
\phi(c_1a_1 + \dots + c_na_n) = 0
$$
and you can probably finish it from here.
