PS3 - Mathematic 104 Fall 2010 Assignment#3 Due Wednesday...

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Mathematic 104, Fall 2010: Assignment #3 Due: Wednesday, October 20th Instructions: Please ensure that your answers are legible. Also make sure that all steps are shown – even for problems consisting of a numerical answer. Bonus problems cover advanced material and, while good practice, are not required and will not be graded. Problem #1. Excercise 2.6 of Lecture 2 of Trefethen-Bau. Problem #2. Excercise 6.3 of Lecture 6 of Trefethen-Bau. Problem #3. Let E be the space in R 3 spanned by 1 0 1 and 0 - 1 1 a) Find the matrix P R 3 × 3 corresponding to orthogonal projection onto E . b) Find a unit vector q R 3 so that P is the complementary projector to P q . Recall P q = qq is the projector onto the space spanned by q . Problem #4. Let A C m × n matrix and B
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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 ´ )....
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