OrthogonalProjections-Solutions

OrthogonalProjections-Solutions - Math 331 - Orthogonal...

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Unformatted text preview: Math 331 - Orthogonal Projections Worksheet - Solutions Here are some Practice problems on finding the standard matrix of an orthogonal projection, 1. Let L be the line thru the origin in R 2 that is parallel to the vector 3 4 . (a) Find the standard matrix of the orthogonal projection onto L . Solution: Method 1: We know that proj L : R 2 R 2 is a linear transformation, so we can find the columns of the standard matrix by plugging in the standard basis vectors. We have the formula proj L ( v ) = v u u u u We have e 1 u = 1 3 4 = 3 e 2 u = 1 3 4 = 4 u u = 3 4 3 4 = 25 So proj L ( e 1 ) = 3 25 3 4 = 9 25 12 25 proj L ( e 2 ) = 4 25 3 4 = 12 25 16 25 Then the standard matrix is A = 9 25 12 25 12 25 16 25 Method 2: We can also use the formula which says that A = UU T where U is the matrix whose columns form an orthonormal basis of L . In the case of a line, an orthonormal basis is just a vector with length one in the direction of L . To find this we just normalize u . u | u | = 1 5 3 4 = 3 5 4 5 So we have U = 3 5 4 5 and thus A = UU T = 3 5 4 5 3 5 4 5 = 9 25 12 25 12 25 16 25 (b) Find the point on L which is closest to the point (7 , 1) and find the point on L which is closest to the point (- 3 , 5)....
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OrthogonalProjections-Solutions - Math 331 - Orthogonal...

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