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Unformatted text preview: R m . 1 5. Let T : R n → R m be a linear map. (a) Expand x in the standard basis e j and show that k T ( x ) k ≤ n X j =1 k T ( e j ) k x j  (b) Apply the Cauchy Schwarz inequality to the estimate in (a) to obtain k T ( x ) k ≤ ± n X j =1 k T ( e j ) k 2 ² 1 2 k x k . 6. Let ϕ ∈ [0 , 2 π ) and deﬁne a map T ϕ : R 2 → R 2 by T ϕ ( x,y ) = ( Re[e i ϕ ( x + i y )] , Im[e i ϕ ( x + i y )] ) , where Re z and Im z stand for the real and imaginary part of the complex number z , respectively. (a) Show that T ϕ is a linear map and compute the associated matrix. (b) By rewriting x + i y in polar form, show that the map T ϕ rotates every vector by an angle ϕ . 2...
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 Spring '10
 Spyros
 Linear Algebra, Algebra, Geometry, Linear map

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