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# Adv Alegbra HW Solutions 124 - 124 Exercises for Chapter 4...

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124 Exercises for Chapter 4 4.1 True or false with reasons. (i) If k is a fi eld, then the subset E of all all polynomials of odd degree is a subspace of k [ x ] . Solution. True. (ii) If A and B are n × n matrices over a fi eld k , and if the homoge- neous system Ax = 0 has a nontrivial solution, then the homoge- neous system ( B A ) x = 0 has a nontrivial solution. Solution. True. (iii) If A and B are n × n matrices over a fi eld k , and if the homoge- neous system Ax = 0 has a nontrivial solution, then the homoge- neous system ( AB ) x = 0 has a nontrivial solution. Solution. False. (iv) If v 1 , v 2 , v 3 , v 4 spans a vector space V , then dim ( V ) = 4. Solution. False. (v) If k is a fi eld, then the list 1 , x , x 2 , . . . , x 100 is linearly indepen- dent in k [ x ] . Solution. True. (vi) There is a linearly independent list of 4 matrices in Mat 2 ( R ) . Solution. True. (vii)
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