# a4f08s - HINTS TO ASSIGNMENT#4 1 The linear map T R 2 R 2...

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HINTS TO ASSIGNMENT #4 1. The linear map is defined by 2 2 : R R T x x T A = ) ( where A is a 2 2 × matrix and x is a column vector. The parametric equation of the line might be written as . v a x t + = Applying T we get the image w b v a v a x T t tA A t A + = + = + = ) ( ) ( which is again the line. We have used the linearity property. Now, a parallelogram may be written as the set of points 1 , 0 , + + = t s t s w v a x where v and w are linearly independent vectors. By linearity it will be mapped to w v b x T tA sA + + = ) ( where vectors A v and A w should be linearly independent to define a parallelogram. This happens (Linear Algebra!) if and only if A is a nonsingular matrix (i.e. det A 0). 2. There are many methods to find a matrix for T (vectors onto vectors condition, or line segments onto line segments condition or vertices onto vertices). Probably the most straightforward would be to use linear algebra. We know that T is determined if we know the action of T on a basis for 2 R and vectors (1,0) and (0,-1) constitute such a basis. We

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a4f08s - HINTS TO ASSIGNMENT#4 1 The linear map T R 2 R 2...

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