One more question:
Let \(\displaystyle Pn(R)\) be the vector space of \(\displaystyle n*n\) matrices over \(\displaystyle R\) and let \(\displaystyle U\) be the set of symmetric \(\displaystyle n*n\) matrices , while \(\displaystyle V\) is the set of antisymmetric \(\displaystyle n*n\) matrices.
(i) Show that \(\displaystyle U\) and \(\displaystyle V\) are subspaces of \(\displaystyle Pn(R)\)
(ii) Show that \(\displaystyle Pn(R)\)=\(\displaystyle U\)\(\displaystyle \bigoplus\) \(\displaystyle V\)
Let \(\displaystyle Pn(R)\) be the vector space of \(\displaystyle n*n\) matrices over \(\displaystyle R\) and let \(\displaystyle U\) be the set of symmetric \(\displaystyle n*n\) matrices , while \(\displaystyle V\) is the set of antisymmetric \(\displaystyle n*n\) matrices.
(i) Show that \(\displaystyle U\) and \(\displaystyle V\) are subspaces of \(\displaystyle Pn(R)\)
(ii) Show that \(\displaystyle Pn(R)\)=\(\displaystyle U\)\(\displaystyle \bigoplus\) \(\displaystyle V\)