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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 1 2 k x k . 6. Let [0 , 2 ) and dene a map T : R 2 R 2 by T ( x,y ) = ( Re[e i ( x + i y )] , Im[e i ( 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 is a linear map and compute the associated matrix. (b) By rewriting x + i y in polar form, show that the map T rotates every vector by an angle . 2...
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