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HWSolutions1.8&amp;2.3

HWSolutions1.8&amp;2.3 - Newberger Math 247 Spring 03...

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Newberger Math 247 Spring 03 Homework solutions: Section 1.8 #26, 27, 31, Section 2.3 #34,36,37,38 Section 1.8 #26,27,31. 26. Let u and v be linearly independent vectors in R 3 and let P be the plane through u , v and 0 . The parametric equation of P is x = s u + t v (with s and t in R ). Show that a linear transformation T : R 3 R 3 maps P onto a plane through 0 , a line through 0 or onto just the origin 0 in R 3 . What must be true about T ( u ) and T ( v ) in order for the image of the plane P to be a plane? We want to know the image of P under T . Any point x on P will have the form x = s u + t v where s and t are real numbers. So T ( x ) = sT ( u ) + tT ( v ) , and T ( P ) is the collection of all vectors of the form sT ( u )+ tT ( v ) where s and t are any real number. Thus the image of P under T is Span { T ( u ) ,T ( v ) } . Since there are three cases for the geometric description of the span of two vectors in R 3 , there are three cases for the geometric description of the image of P under T . First { T ( u ) ,T ( v ) } could be independent, in which case, the

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HWSolutions1.8&amp;2.3 - Newberger Math 247 Spring 03...

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