Exercise 23

We can ask about the algebra of the ‘$+$’ operation.

• Is it commutative; is $W_{1}+W_{2}=W_{2}+W_{1}$?

• Is it associative; is $(W_{1}+W_{2})+W_{3}=W_{1}+(W_{2}+W_{3})$?

• Let $W$ be a subspace of some vector space. Show that $W+W=W$.

• Must there be an identity element, a subspace $I$ such that $I+W=W+I=W$ for all subspaces $W$?

• Does left-cancellation hold: if $W_{1}+W_{2}=W_{1}+W_{3}$ then $W_{2}=W_{3}$? Right cancellation?