I am currently working on two algebra questions that ask me to answer the following questions on matrix $A$, a $m \times n$ matrix given the following conditions:

a) $n > m$

b) $m > n$

1) There always exists an $x$ not equal to zero such that $Ax = 0$

2) It is possible for the columns of $A$ to span $\mathbb{R}^m$

For question (1) my plan was to provide an example proving the statement or a reason why it is not true, but I am really struggling to find any combinations of $A$ and $x$ that equal zero given the two conditions. 

And for the second question I think I am completely missing the point of this question, because I don't see why it would not be possible for the columns of $A$ to span $\mathbb{R}^m$.