MidSol6 - 1 w 2 = 1 5 . The general formula for the matrix...

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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 = 1 5 . The general formula for the matrix of the orthogonal projection onto the line spanned by w is 1 w 2 1 + w 2 2 w 2 1 w 1 w 2 w 1 w 2 w 2 2 , so in our particular case the matrix becomes 1 26 1 5 5 25 = 1 26 5 26 5 26 25 26 . This can also be derived using the formula for the orthogonal projection: proj w ( v ) = 1 | w | 2 ( v w ) w . V. K. 6...
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