mat67-2011-hw4solutions

mat67-2011-hw4solutions - Solutions to HW#4 Math 67 UC...

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Solutions to HW #4 Math 67 UC Davis, Fall 2011 1. Solve the following problems in the textbook: (a) Proof-writing exercise 6 in Chapter 5. Solution. Since U + V R 9 , we have dim( U + V ) 9, and therefore, by theorem 5.4.6 in the textbook, dim( U V ) = dim( U )+dim( V ) - dim( U + V ) dim( U )+dim( V ) - 9 = 5+5 - 9 = 1 . So U V must be larger than the 0-dimensional space { 0 } . (b) Calculational exercises 1(a),(b),(c),(f), 2(a)–(b), 5, 6 in Chapter 6. Solution to 1(b),(c). For each ( a,b ) R 2 , it is easy to find that the equation ( x + y,x ) = T ( x,y ) = ( a,b ) has the unique solution ( x,y ) = ( b,a - b ). Since there is a solution, that means the transformation is surjective. Since the solution is unique, that means the transformation is injective, which as we saw is equivalent to null( T ) = { 0 } , so dim(null( T )) = 0. Solution to 1(f). F (0 , 0) = (0 , 1). Since F does not map the zero vector in the domain to the zero vector in the codomain, it is not a linear transformation. (c) Proof-writing exercises 2, 3 in Chapter 6.
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mat67-2011-hw4solutions - Solutions to HW#4 Math 67 UC...

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