Straight Sum: Let Pn(R) be the vector space of n∗n matrices over R and let U be the set of symmetric n∗n matrices, while V is...

[spinos]

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 \(\displaystyle Pn(R)\)=\(\displaystyle U\)\(\displaystyle \bigoplus\) \(\displaystyle V\)

 

[khansaheb]

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 \(\displaystyle Pn(R)\)=\(\displaystyle U\)\(\displaystyle \bigoplus\) \(\displaystyle V\)

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