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Unformatted text preview: Remainder: Next time there will be a quiz. It will cover the material of chapters 1.12.3 ( including 2.3). You’ll have 10 or 15 minutes depending on the difficulty of the problems (I haven’t decided yet.) Format: closed book & no calculators. ================================================================= MATH 294 Chapter 2.2 Linear Transformations in Geometry Ex 1 : Suppose a line L in R n is spanned by a unit vector ~u = ( u 1 , . . . , u n ) t , that is, L = { λ~u, λ ∈ R } and the length of ~u , k u k = p u 2 1 + . . . + u 2 n = 1. Find the matrix A of the linear transformation T ( ~x ) = proj L ~x . Give the entries of A in terms of the components u i of ~u . Solution: First of all, let’s notice that ~u is fixed so its components are fixed scalars. From the fact 2.2.5 we know that an orthogonal projection is a linear transformation. Hence, there exists a matrix A such that T ( ~x ) = A~x . The same fact says that T ( ~x ) = proj L ~x = ( ~u · ~x ) ~u = ( ∑ n i =1 u i x i ) u 1 ....
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 Fall '05
 HUI
 Math, Linear Algebra, Remainder, 15 minutes, 1 K, linear transformation, 0 1 1 0 k

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