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lect136_8_w09

# lect136_8_w09 - Friday January 16 Lecture 8 Linear...

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Friday January 16 Lecture 8 : Linear operators on R n . (Refers to section 3.3 ) Expectations : 1. Define linear operator. 2. Recognize linear operators. 3. Apply simple linear operators 4. Show projections are linear operators. 8.1 Definition A linear transformation mapping R n into R n is called a linear operator . 8.1.1 Familiar examples of linear operators on R 2 : Rotation of a vector counterclockwise about the origin by an angle of θ . For a chosen value of θ we have the shown that the matrix induced by this linear transformation is: cos θ sin θ sin θ cos θ Reflection transformation about the x-axis .The matrix which is induced by this linear transformation was shown to be 1 0 0 1 since 1 0 x x 0 1 y = y 8.2 We study some more linear operators on R 2 as well as on R 3 . 8.2.1 The projection of a vector b onto a vector a is given by the expression

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p = proj a ( b ) = [ < a , b > ] ( a / || a || 2 ) . 8.2.1.1 Example Consider the vector b = ( 4, 3) and let a = (2, 0). It is easy, in this case, to graphically visualize, and then compute what proj a ( b ) is. Then proj a ( b ) = 2(2, 0) = ( 4, 0) and w = ( 4, 3) ( 4, 0) = (0, 3). 8.2.1.2 ( This proof is optional material although one is expected to understand the flow of it. However, one is expected to know by heart the formula for proj a ( b ).) Proof of a formula for the projection p = proj a ( b ) of b onto a . ( Optional ) By the definition the points b , p = t a and w = b p form a right triangle in a plane ( b forming the hypotenuse), so < p , b p > = 0. Recall that a / || a || is the unit vector in the direction of a . Since p = t a , then a / || a || = ± p / || p || .
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