lina-20-11-09-p1 - R m 1 5 Let T R n → R m be a linear...

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Linear Algebra & Geometry: Sheet 7 Set on Friday, November 20: Questions 1, 2, and 3 1. Determine which of the following maps are linear maps (a) T : R 2 R 2 , T ( x, y ) = ( x - y, 5 x ). (b) T : R 2 R 2 , T ( x, y ) = ( x - y 2 , 5 x ). (c) T : R 2 R 2 , T ( x, y ) = ( x - 1 , 5 x ). (d) T : R 2 R 2 given by T ( x ) = ( x · e 1 ) u 1 + ( x · e 2 ) u 2 where u 1 = ( - 1 , 3) and u 2 = (3 , 3) (and e 1 = (1 , 0) and e 2 = (0 , 1)). (e) T : R 2 R , T ( x, y ) = e x - cos( πy ). Remark : in Problems 1 and 2 we use the notation ( x 1 , x 2 ) to denote the vector in R 2 with first component x 1 and second component x 2 . 2. Compute the matrix associated with a linear map for the following maps: (a) T : R 2 R 2 given by T ( x, y ) = ( x - y, 5 x ). (b) T : R 2 R 2 given by T ( x ) = ( x · e 1 ) u 1 + ( x · e 2 ) u 2 where u 1 = ( - 1 , 3) and u 2 = (3 , 3) (and e 1 = (1 , 0) and e 2 = (0 , 1)). . (c) T : R 3 R given by T ( x, y, z ) = x - y + 5 z . (d) T : R R 3 given by T ( x ) = ( x, - x, 11 x ). 3. Compute A x for (a) A = 0 1 - 1 0 and x = e 1 and x = e 2 . (b) A = - 1 0 0 1 and x = e 1 and x = e 2 . (c) A = 0 1 3 2 - 1 0 and x = 1 3 9 . (d) A = 0 1 3 2 - 1 0 and x = 1 9 . 4. Let T : R n R m be a linear map and V R n a linear subspace. Show that T ( V ) := { y R m ; there exist a x R n with T ( x ) = y } is a linear subspace of
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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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