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204PS42008

# 204PS42008 - Econ 204 Summer/Fall 2008 Problem Set 4 Due in...

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Econ 204, Summer/Fall 2008 Problem Set 4 Due in Lecture Tuesday, August 12 1. Show that each of the following is a vector space over R , find a Hamel basis, and the dimension of the space: (a) The set of solutions in R 3 to the following systems of linear equations x 1 2 x 2 + x 3 = 0 2 x 1 3 x 2 + x 3 = 0 (b) The set of all n × n matrices having trace 1 equal to zero. 2. T : M 2 × 3 M 2 × 2 defined by T a 11 a 12 a 13 a 21 a 22 a 23 = 2 a 11 a 12 a 13 + 2 a 12 0 0 Determine Ker ( T ), dim( Ker ( T )) and Rank ( T ). Is T one-to-one, onto, or neither? 3. Derive a transformation, T : R 2 R 2 , which reflects a point across the line y = 5 x . (a) First, calculate the action of T on the points (1 , 5) and ( 5 , 1). (b) Next, write the matrix representation of T using these two vectors as a basis. (c) Find S and S - 1 , the matrices that changes coordinates under this basis to standard coordinates and back again. (d) Write the matrix representation of T in the standard basis.

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