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midterm_1_sol - 2.035 Midterm Exam Part 1 Spring 2007...

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2.035: Midterm Exam - Part 1 Spring 2007 SOLUTION PROBLEM 1: a) A vector space is a set V of elements called vectors together with operations of addition and multiplication by a scalar, where these operations must have the following properties: (A) Corresponding to every pair of vectors x , y V there is a vector in V , denoted by x + y , and called the sum of x and y , with the following properties: (1) x + y = y + x for all x , y V ; (2) x + ( y + z ) = ( x + y ) + z for all x , y , z V ; (3) there is a unique vector in V , denoted by o and called the null vector, with the property that x + o = x for all x V ; and (4) corresponding to every vector x V there is a unique vector in V , denoted by x with the property that x + ( x ) = o . (B) Corresponding to every real number α R and every vector x V there is a vector in V , denoted by α x , and called the product of α and x , with the following properties: (5) α ( β x ) = ( αβ ) x for all α, β R and all x V ; (6) α ( x + y ) = α x + α y for all α R and all x , y V ; (7) ( α + β ) x = α x + β x for all α, β R and all x V ; and (8) 1 x = x for all x V . b) A set of vectors { f 1 , f 2 , . . . , f n } is said to be linearly independent if the only scalars α 1 , α 2 , . . . , α n for which α 1 f 1 + α 2 f 2 . . . + α n f n = o are α 1 = α 2 = . . . = α n = 0. c) If a vector space V contains a linearly independent set of n ( > 0) vectors but contains no linearly independent set of n + 1 vectors we say that the dimension of V is n . d) If V is a n -dimensional vector space then any set of n linearly independent vectors is called a basis for V . e) If { f 1 , f 2 , . . . , f n } is a basis for an n -dimensional vector space V , then any vector x V can be expressed in the form x = ξ 1 f 1 + ξ 2 f 2 + . . . + ξ n f n where the set of scalars ξ 1 , ξ 2 , . . . , ξ n is unique and are called the components of x in the basis { f 1 , f 2 , . . . , f n } .
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f) To every pair of vectors x , y V we associate a real number denoted by x y and called the · scalar product of x and y provided that this product has the following properties: (9) x y = y x for
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