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Unformatted text preview: HOMEWORK ASSIGNMENT 2: SOLUTIONS TO THE GRADED EXERCISES MATH 115A, SECTION 3, FALL 2011 Ex. 11, Section 1.4 (*) Solution. We must prove that both sets are equal, proving that each one is contained in the other one. ) . Let v P span ( t x u ) . By definition, this means that v is a linear combination over F of the elements of t x u . Since this set only has one element, any linear combination must be of the form v = ax for some a P F , proving the first inclusion. ) . Let v P t ax : a P F u . Then v = ax for some a P F , which is a linear combination of the elements of t x u , proving that v P span ( t x u ) . In R 2 , geometrically this means that the span of any non-zero vector x is the line passing through x and 0. The span of the zero vector consists just of the origin. Ex. 7, Section 1.5 (*) Solution. Any 2 2 diagonal matrix with coefficients in F is of the form A = a 1 a 2 , with a 1 , a 2 P F . By definition of the operations on M 2 2 ( F ) , this can be written as A = a 1 1 + a 2 1 . Thus, the set " 1 , 1 * is a set of generators for M 2 2 ( F ) , since we just proved that any diagonal matrix is a linear combination of these two matrices. It is also linearly independent, for suppose we have a linear combination of these two elements equal to 0. That is, let a 1 , a 2 P F be such that a 1 1 + a 2 1 = . Then a 1 a 2 = , and by definition of equality of two matrices, we must have a 1 = a 2 = 0. This proves the linear independence....
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