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Unformatted text preview: 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.- 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 ....
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