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Unformatted text preview: Partial Solution Set, Leon 3.1 3.1.3 We are to show that the set C of complex numbers, with scalar multiplication defined by ( a + bi ) = a + bi and addition defined by ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d ) i , satisfies the eight axioms of a vector space. This is only a partial solution. A1: Let a + bi,c + di C . Then ( a + bi ) + ( c + di ) = (( a + c ) + ( b + d ) i ) (By definition of complex addition) = (( c + a ) + ( d + b ) i ) (Real addition is commutative) = ( c + di ) + ( a + bi ) (By definition of complex addition) A2: Similar to A1; pick three complex numbers, use the definition of complex addition as often as necessary, together with the known associativity of real addition, to show that complex addition is associative. A3: The zero element is = (0 + 0 i ). A4: To show existence of the additive inverse, choose an arbitrary complex number (say, x = a + bi ) and construct its additive inverse. This will be made easy by your knowledge of real additive inverses. A5: We must prove that scalar multiplication distributes over complex addition. Let a + bi,c + di C , and let R . Then (( a + bi ) + ( c + di )) = (( a + c ) + ( b + d ) i ) (Defn complex addition) = ( a + c ) + ( b + d ) i (Defn of scalar mult. in C ) = ( a + c ) + ( b + d ) i (Distributivity in R ) = ( a + bi ) + ( c + di...
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