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1.2(8,12) 1.3 - Linear Algebra-115 Solutions to First...

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Linear Algebra -115 Solutions to First Homework Problem 8 (Section 1 . 2) We have: ( a + b )( x + y ) = a ( x + y ) + b ( x + y ) , because of axiom VS8 = ax + ay + bx + by, because of axiom VS7 Problem 12 (Section 1 . 2) First we observe that the set of even functions (call it Even) is a subset of F ( S, F ). By Theorem 1 . 3 it is a subspace iff three conditions are met. We are verifying them one by one: 1. -→ 0 Even . Note here that the zero vector is actually the zero function, i.e. the one that gives value zero for all arguments. Call this function z ( x ). It is even, since z ( x ) = 0 = z ( - x ). 2. Say f, g Even . Then f ( x ) = f ( - x ) , g ( x ) = g ( - x ). Adding these up we get f ( x )+ g ( x ) = f ( - x )+ g ( - x ), or in other words f + g ( x ) = f + g ( - x ), by the definition of f + g . But this is like saying the f + g Even . 3. Similarly to the above, multiply f ( x ) = f ( - x ) by c , to get cf ( x ) = cf ( - x ). From this we conclude that cf Even . Now, since Even is a subset of F ( S, F ), it has to be a vector subspace by itself. Note: There is another way to go for the problem. Verify all 8 axioms and the fact that Even is closed for addition and multiplication. This way is correct,
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