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 nonnegative 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 Fdim( 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 ´ )....
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 '09
 Math, Linear Algebra, orthogonal projector Pv

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