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1. (Artin 3.4.1) Find a basis for the space of n×n symmetric matrices (those for which At = A). 2. (Artin 3.4.2) Let W ⊂ R 4 be the space of solutions of the system of linear equations AX = 0, where 2 1 2 3 1 1 3 0 . Find a basis for W. 3. (Artin 3.4.4) Let A be an m × n matrix, and let A0 be the result of a sequence of elementary row operations on A. Prove that the rows of A span the same space as the rows of A0 . 4. (Artin 3.5.1) (a) Prove that the (ordered) set B = (1, 2, 0)t ,(2, 1, 2)t ,(3, 1, 1)t  is a basis of R 3 . (b) Find the coordinate vector of the vector v = (1, 2, 3)t with respect to this basis. (c) Let B 0 = (0, 1, 0)t ,(1, 0, 1)t ,(2, 1, 0)t  . Determine the change-of-basis matrix P from B to B0 . 5. (Artin 3.5.4) Let Fp be the field of p elements for some prime p, and let V = F 2 p . Prove: (a) The number of bases of V is equal to the order of GL2(Fp). (b) The order (i.e. size) of GL2(Fp) is p(p + 1)(p − 1)2 , and the order of SL2(Fp) is p(p + 1)(p − 1). (Recall that SL2(F) is the subgroup of GL2(F) of matrices over a field F having determinant 1.) 6. (Artin 3.5.5) How many subspaces of each dimension are there in (a) F 3 p ? (b) F 4 p ?

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