MidSol6

# MidSol6 - 1 w 2 Â = Âµ 1 5 Â The general formula for the...

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9. Let −→ v 1 , −→ v 2 , −→ v 3 be columns of A , i.e. A =[ −→ v 1 −→ v 2 −→ v 3 ]. Since A 0 0 1 is just −→ v 3 , from the Frst equation we get −→ v 3 = 2 1 0 . After that, since A 3 0 1 =3 −→ v 1 + −→ v 3 , we obtain from the second equation −→ v 1 = 1 3 0 0 1 1 3 −→ v 3 = 1 3 0 0 1 1 3 2 1 0 = 2 3 1 3 1 3 . ±inally, from A 2 1 0 =2 −→ v 1 + −→ v 2 , and the third equation we get: −→ v 2 = 3 0 1 2 −→ v 1 = 3 0 1 2 2 3 1 3 1 3 = 13 3 2 3 1 3 . Therefore A = 2 3 13 3 2 1 3 2 3 1 1 3 1 3 0 . 10. The line y =5 x is spanned (determined) for instance by the vector −→ w = µ w
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Unformatted text preview: 1 w 2 Â¶ = Âµ 1 5 Â¶ . The general formula for the matrix of the orthogonal projection onto the line spanned by âˆ’â†’ w is 1 w 2 1 + w 2 2 Âµ w 2 1 w 1 w 2 w 1 w 2 w 2 2 Â¶ , so in our particular case the matrix becomes 1 26 Âµ 1 5 5 25 Â¶ = Âµ 1 26 5 26 5 26 25 26 Â¶ . This can also be derived using the formula for the orthogonal projection: proj âˆ’â†’ w ( âˆ’â†’ v ) = 1 | âˆ’â†’ w | 2 ( âˆ’â†’ v Â· âˆ’â†’ w ) âˆ’â†’ w . V. K. 6...
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## This note was uploaded on 04/12/2011 for the course MATH 33a taught by Professor Lee during the Fall '08 term at UCLA.

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