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Unformatted text preview: 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 ndimensional vector space then any set of n linearly independent vectors is called a basis for V ....
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.035 taught by Professor Rohanabeyaratne during the Spring '07 term at MIT.
 Spring '07
 RohanAbeyaratne

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