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

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Remainder: Next time there will be a quiz. It will cover the material of chapters 1.1-2.3 ( 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 = { λu, λ R } and the length of u , u = 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 ) = Ax . The same fact says that T ( x ) = proj L x = ( u · x ) u = ( n i =1 u i x i ) u 1 . . . u n = ( n i =1 u i x i ) u 1 . . . ( n i =1 u i x i ) u n = u 1 x 1 u 1 + u 1 x 2 u 2 + . . . + u 1 x n - 1 u n - 1 + u 1 x n
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