linalg.proj(1)

linalg.proj(1) - Projections on a line : LinearAlg Jonathan...

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Projections on a line : LinearAlg Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.uﬂ.edu/ squash/ 20 November, 2006 (at 18:39 ) Problem Let P = P 30 be the 2 × 2 matrix whose lefthand action is ortho-projection on the angle=30 line through the origin. Compute P . Support your answer in three conceptually-di±erent ways. Strategy Step S 0 . A simplifying notation: Use c := cos(30 ) and s := sin(30 ). Step S 1 . Use similar triangles to compute P · ˇ e 1 ; this product gives the ﬁrst column of P . Use sim.tri to compute P · ˇ e 2 ; this equals the second column of P . Step S 2 . Checking our result: Each vector v on the 30 line should be ﬁxed (not moved) by the pro- jection. So P · ˇ v better equal ˇ v . Step S 3 . “Variation of parameters”: Generalize the result of Step1, replacing “30 ” by a general an- gle ω . Easily P 0 = " 1 0 0 0 # and P 90 = " 0 0 0 1 # . 1 : Now we will check that our
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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.

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