2題 Basic Matrix Transfortion

1. (a) A matrix is said to be antisymmetric if
圖片參考：http://imgcld.yimg.com/8/n/HA00425268/o/701102010132113873376340.jpg
. Show that a general 2 x 2 antisymmetric matrix is of the form

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(b) Let the set of 2 x 2 antisymmetric matrices be denoted S. By making a correspondence between elements of M2;2 and four-tuples (a; b; c; d), write the subset of R^4 corresponding to S as a span. What is the dimension of this subspace?
(c) Show from first principles that the set S is closed under scalar multiplication.


2. Let the standard matrix representation of a linear transformation T : R^2 → R^2 be given by

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(a) Compute the image of the unit vectors e1 = (1; 0), e2 = (0; 1).(b) Compute the image of the unit square {(x; y) : 0 ≦ x ≦ 1; 0 ≦ y ≦ 1}.(c) Describe T geometrically.