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**Unformatted text preview: **C n k matrix. Show that a necessary and sucient condition for R ( AB ) = R ( A ) is that N ( A ) + R ( B ) = C n . Problem #5. Let v = cos sin R 2 . a) Find the orthogonal projector P v R 2 2 . b) Find the orthogonal matrix U R 2 2 so that P v = UXU > where X is a diagonal matrix with non-negative entries on the diagonal. (Hint: Think geometrically). Bonus Problem. Let E,F C n be two vector spaces. Show that dim( E + F ) = dim E + dim F-dim( E F ) . (Hint: Let v 1 ,...,v k be a basis E and w 1 ,...,w l be a basis of F . Consider the n ( k + l ) matrix X = v 1 | | v k | w 1 | | w l )....

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