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Linear_Algebra_Notes_2 vector spaces

Linear_Algebra_Notes_2 vector spaces - Linear Algebra Notes...

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Linear Algebra, Notes 2 Def 4.4 A real vector space is a set V of elements on which we have two operations + and * defined with these properties: (a) if u, v are elements in V , then u+v is in V (closed under +): (i) u+v = v+u for all u, v in V (ii) u+(v+w) = (u+v)+w for u, v, w in V (iii) there exists an element 0 in V such that u+0 = 0+u = 0 for u in V. (iv) for each u in V there exists an element -u in V such that u+(-u) = (-u) +u = 0. (b) If u is any elemnt in V and c is a real number, then c*u (or cu) is in V (V is closed under scalar multiplication). (i) c*(u+v) = c*u + c*v for any u,v in V, c a real number (ii) (c+d) * u = c*u + d*u for any u in V, c, d real numbers (iii) c * (d*u) = (cd) * u for any u in V, c, d real numbers (iv) 1*u = u for any u in V Theorem 4.2 If V is a vector space, then (a) 0*u = u for any u in V (b) c*0 = 0 for any scalar c (c) if c*u = 0, then either c = 0 or u = 0. (d) (-1)*u = -u for any vector u in V Def 4.5 Let V be a vector space and W a nonempty subset of V. If W is a vector space with respect to the operations in V, then W is called a subspace of V. Theorem 4.3 Let V be a vector space and let W be a nonempty subset of V. Then W is a subspace of V if and only if the following conditions hold: (a) if u, v are in W, then u+v is in W (b) if c is a real number and u is any vector in W, then c*u is in W. Definition 4.6 Let v 1 , v 2 , ..., v n be vectors in vector space V. A vector V is a linear combination of v 1 , v 2 , ..., v n , if v = a 1 v 1 + a 2 v 2 + ... + a n v n Definition 4.7 If S = {v 1 , v 2 , ..., v n

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Linear_Algebra_Notes_2 vector spaces - Linear Algebra Notes...

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