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Unformatted text preview: 2.035: Midterm Exam Part 2 Spring 2007 SOLUTION Problem 1: Consider the set V of all 2 × 2 skewsymmetric matrices, i.e. matrices of the form ⎛ ⎞ x x = ⎝ ⎠ , −∞ < x < ∞ . (1.1) − x a) Define the addition of two vectors by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x + y x y x + y = ⎝ ⎠ where x = ⎝ ⎠ , y = ⎝ ⎠ , (1.2) − x − y − x − y and the multiplication of a vector x by the scalar α as ⎛ ⎞ αx α x = ⎝ ⎠ . (1.3) − αx Then one can readily verify that x + y ∈ V whenever x and y ∈ V , and that α x ∈ V for every scalar α whenever x ∈ V . Thus V is a vector space. b) Since ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x y y ⎝ ⎠ − x ⎝ ⎠ = ⎝ ⎠ (1.4) − x − y it follows that y x + ( − x ) y = o for any two vectors x and y . Thus every pair of vectors is linearly dependent. There is no pair of vectors that is linearly independent. c) It follows from item (b) that the dimension of the vector space is 1. d) If x and y are two vectors in V , and we define x y = xy + ( − x )( − y ) = 2 xy, (1.5) · then we can verify that x y has the properties · a) x y = y x , · · b) x ( α y + β z ) = α x y + β x z , · · · c) x x ≥ with x x = if and only if x = o , · · for all x , y , z ∈ V and all scalars α, β . Thus this is a proper definition of a scalar product. e) Since the dimension of V is 1, any one (non null) vector provides a basis for it. Pick ⎛ ⎞ 1 e = ⎝ ⎠ (1.6) − 1 Note that the length of e is  e  = √ e · e = √ 1 + 1 = √ 2 . (1.7) Thus a basis composed of a unit vector is { e } where ⎛ ⎞ 1 / √ 2 e = ⎝ ⎠ . (1.8) − 1 / √ 2 f) Let A be the transformation defined by ⎛ ⎞ 2 x Ax = ⎝ ⎠ for all vectors x ∈ V . (1.9) − 2 x We can readily verify that ( i ) the vector Ax ∈ V for every x ∈ V , and ( ii ) that A ( α x + β y ) = α Ax + β Ay for all vectors x , y and all scalars α, β . Therefore A is a linear transformation (tensor)....
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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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