OrthogonalProjections

# - P(b Find the standard matrix of the orthogonal projection onto P(c Find the point on P which is closest to the point(1 0 4 Let x 1 = 1 and x 2 =

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Math 331 - Orthogonal Projections Worksheet 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 ² . (a) Find the standard matrix of the orthogonal projection onto L . (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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2. Let L be the line thru the origin in R 3 which is parallel to the vector 1 - 1 2 . (a) Find the standard matrix of the orthogonal projection onto L . (b) Find the point on L which is closest to the point (1 , 0 , 0).
3. Let x 1 = 1 2 1 and x 2 = 3 0 3 and let P be the plane thru the origin spanned by x 1 and x 2 . (a) Find an orthonormal basis of

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Unformatted text preview: P . (b) Find the standard matrix of the orthogonal projection onto P . (c) Find the point on P which is closest to the point (1 , , 0), 4. Let x 1 = 1 and x 2 = 1 1 1 and let P be the plane thru the origin spanned by x 1 and x 2 . (a) Find an orthonormal basis of P . (b) Find the standard matrix of the orthogonal projection onto P . (c) Find the point on P which is closest to the point (0 , , 1), 5. Describe the column space and null space of each of the matrices you found in problems 1-4. (You do not need to do any row operations to answer this question)...
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- P(b Find the standard matrix of the orthogonal projection onto P(c Find the point on P which is closest to the point(1 0 4 Let x 1 = 1 and x 2 =

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