book (dragged) 11 - Chapter Two Vector Spaces 92(b This set...

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92 Chapter Two. Vector Spaces (b) This set { 0 a b 0 | a, b 2 C and a + b = 0 + 0i } 1.43 Name a property shared by all of the R n ’s but not listed as a requirement for a vector space. X 1.44 (a) Prove that for any four vectors ~ v 1 , . . . , ~ v 4 2 V we can associate their sum in any way without changing the result. (( ~ v 1 + ~ v 2 ) + ~ v 3 ) + ~ v 4 = ( ~ v 1 + ( ~ v 2 + ~ v 3 )) + ~ v 4 = ( ~ v 1 + ~ v 2 ) + ( ~ v 3 + ~ v 4 ) = ~ v 1 + (( ~ v 2 + ~ v 3 ) + ~ v 4 ) = ~ v 1 + ( ~ v 2 + ( ~ v 3 + ~ v 4 )) This allows us to write ‘ ~ v 1 + ~ v 2 + ~ v 3 + ~ v 4 ’ without ambiguity. (b) Prove that any two ways of associating a sum of any number of vectors give the same sum. ( Hint. Use induction on the number of vectors.) 1.45 Example 1.5 gives a subset of R 2 that is not a vector space, under the obvious operations, because while it is closed under addition, it is not closed under scalar multiplication. Consider the set of vectors in the plane whose components have the same sign or are 0 . Show that this set is closed under scalar multiplication but not addition.
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