notes6-2

# notes6-2 - of u and another vector z that is orthogonal to...

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SECTION 6.2 ORTHOGONAL SETS ORTHOGONAL SET OF VECTORS. Each pair of distinct vectors in the set is or- thogonal. ORTHOGONAL BASIS. A basis that is also an orthogonal set. ORTHONORMAL SET. An orthogonal set of unit vectors. ORTHONORMAL BASIS. A basis that is also an orthonormal set. EXAMPLE. Is the following set orthogonal? 3 3 1 - 2 , 4 - 1 - 3 3 , 0 3 7 8 How would we make this set into an orthonormal set?

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FACT 1. Orthogonal sets of nonzero vectors are linearly independent. FACT 2. If { u 1 , u 2 ,..., u p } is an orthogonal basis for a subspace W of R n and y is in W , then the weights in y = c 1 u 1 + c 2 u 2 + ··· + c p u p are given by c j = y · u j u j · u j .
EXAMPLE. Show that { u 1 , u 2 , u 3 } is an orthogonal basis for R 3 and express x as a linear combination of the u ’s. u 1 = 0 1 1 , u 2 = 4 - 1 1 , u 3 = 1 2 - 2 , x = - 4 8 - 3 ORTHOGONAL PROJECTION. Given a nonzero vector u in R n and any vector y in R n , we often want to write y as the sum of a multiple ˆ y

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Unformatted text preview: of u and another vector z that is orthogonal to u . We usually call ˆ y the projection of y onto u . When c 6 = 0, then ˆ y is the same as the projection of y onto c u . CHECK THIS! As a result we can think of ˆ y as the projection of y onto the subspace (or line) L spanned by u . Consequently we write ˆ y = proj u y = proj L y = y · u u · u u . EXAMPLE. Compute the orthogonal projection of " 3 5 # onto the line through "-1 2 # as well as the distance from " 3 5 # to this line. FACT 3. An m × n matrix U has orthonormal columns exactly when U T U = I . FACT 4. Multiplying vectors by matrices whose columns are orthonormal preserves lengths, dot products, and orthogonality. ORTHOGONAL MATRICES. An orthogonal matrix is a square matrix whose inverse is its transpose. HOMEWORK: SECTION 6.2...
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notes6-2 - of u and another vector z that is orthogonal to...

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