Section 3 Notes - Remainder: Next time there will be a...

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Unformatted text preview: Remainder: Next time there will be a quiz. It will cover the material of chapters 1.1-2.3 ( including 2.3). Youll have 10 or 15 minutes depending on the difficulty of the problems (I havent 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 = { ~u, 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, lets 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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Section 3 Notes - Remainder: Next time there will be a...

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