Math416_HW1_sol - Math 416 - Abstract Linear Algebra Fall...

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Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E13 Homework 1 solutions Section 1.1 1.1. (1 pt check) 2 x = 2 1 2 3 = 2 4 6 . 3 y = 3 y 1 y 2 y 3 = 3 y 1 3 y 2 3 y 3 . x + 2 y- 3 z = 1 2 3 + 2 y 1 y 2 y 3 - 3 4 2 1 = 1 2 3 + 2 y 1 2 y 2 2 y 3 - 12 6 3 = 2 y 1- 11 2 y 2- 4 2 y 3 . 1.2a. (1 pt) C [0 , 1] := { f : [0 , 1] R | f is continuous } is a vector space. First note that C [0 , 1] is closed under addition and scalar multiplication: For any f,g C [0 , 1] and R , the functions f + g and f are continuous. Since addition and scalar multiplication in C [0 , 1] are defined pointwise and R is a vector space, the eight vector space axioms hold for C [0 , 1] as well. For example, let us check associativity of addition explicitly. For any f,g,h C [0 , 1] and x [0 , 1], we have (( f + g ) +...
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Math416_HW1_sol - Math 416 - Abstract Linear Algebra Fall...

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