**Unformatted text preview: **C n × k matrix. Show that a necessary and suﬃcient condition for R ( AB ) = R ( A ) is that N ( A ) + R ( B ) = C n . Problem #5. Let v = ± cos θ sin θ ² ∈ 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 = ³ v 1 | ··· | v k | w 1 | ··· | w l ´ )....

View
Full
Document