Let V = M (R). Show that the symmetric bilinear form : VxVR given by (X,Y)=...

 

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Let V = M (R). Show that the symmetric bilinear form : VxVR given by (X,Y)= tr(XY) is nondegenerate. (Hint. Suppose X = (a) is such that (X,Y)= 0 for all Y EV. Plug in Y = elk the matrix whose (ek)-entry is 1, and all other entries are 0, to show that zij = 0 for all i, j.) Let V = M (R). Show that the symmetric bilinear form : VxVR given by (X,Y)= tr(XY) is nondegenerate. (Hint. Suppose X = (a) is such that (X,Y)= 0 for all Y EV. Plug in Y = elk the matrix whose (ek)-entry is 1, and all other entries are 0, to show that zij = 0 for all i, j.)