==== Let V Mat2x2 (R) be the vector space of 2 x 2 matrices with real...

  

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==== Let V Mat2x2 (R) be the vector space of 2 x 2 matrices with real entries. The transpose of a 2 x 2 matrix 1 = (a 2) A= C is defined by A' = - (* 2) - (i a). = C A matrix A is symmetric if A = A' and skew-symmetric if A=-A'. Consider the subsets U,W CV defined by U = {AEV: A is symmetric} W={AEV: A is skew-symmetric}. (a) Show that U and W are subspaces of V. (b) Show that V=U+W. ==== Let V Mat2x2 (R) be the vector space of 2 x 2 matrices with real entries. The transpose of a 2 x 2 matrix 1 = (a 2) A= C is defined by A' = - (* 2) - (i a). = C A matrix A is symmetric if A = A' and skew-symmetric if A=-A'. Consider the subsets U,W CV defined by U = {AEV: A is symmetric} W={AEV: A is skew-symmetric}. (a) Show that U and W are subspaces of V. (b) Show that V=U+W.