Kernel of a Linear Map on A Tensor Product Suppose I have the linear maps $ l,k: V \otimes V \rightarrow V \otimes V$ defined by
$ l( e_{i_1} \otimes e_{i_2} ) = e_{i_1} \otimes e_{i_2} + e_{i_2} \otimes e_{i_1}$
and 
$ k( e_{i_1} \otimes e_{i_2} ) = e_{i_1} \otimes e_{i_2} - e_{i_2} \otimes e_{i_1}$
where $ (e_{i})_{i=1}^{n} $ is a basis of V.
What are  the kernels of these two maps? 
Or at least an element in both their kernels... other than zero.

Try to describe $l$ and $k$ in coordinate-free ways. How do each act on a $v\otimes w$? Then you can describe them as projections onto complementary subspaces. (Consider the words "symmetric" and "alternating," or in the context of functions, remember what "even" and "odd" parts are.)
