math125-lecture 4.vs - V is closed under the operation is...

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Unformatted text preview: V is closed under the operation is commutative is associative An additive identity exists in V For every element in V, there is an additive inverse A real vector space is a set V of objects together with two operations and satisfying the following properties: (for all u, v, and w in V and real numbers c and d ) Sec 4.VS Saturday, February 06, 2010 11:23 PM Section4VS Page 1 V is closed under the operation is commutative is associative An additive identity exists in V For every element in V, there is an additive inverse Just like for addition of real numbers A real vector space is a set V of objects together with two operations and satisfying the following properties: (for all u, v, and w in V and real numbers c and d ) 4.VS.1 Saturday, February 06, 2010 11:23 PM Section4VS Page 2 A real vector space is a set V of objects together with two operations and satisfying the following properties: (for all u, v, and w in V and real numbers c and d ) V is closed under the operation If u and v are in V, then u v is in V is commutative u v = v u is associative u ( v w ) = ( u v ) w An additive identity exists in V There is an element 0 in V such that u 0 = u For every element in V, there is an additive inverse Given a in V, there is a b in V such that a b = 0. We write b = - a Just like for addition of real numbers → → → → → → → → → → → → → → → → → → → → → → → → → 4.VS.2 Saturday, February 06, 2010 11:23 PM Section4VS Page 3 A real vector space is a set V of objects together with two operations and satisfying the following properties: (for all u, v, and w in V and real numbers c and d ) V is closed under the operation If u and v are in V, then u v is in V is commutative u v = v u is associative u ( v w ) = ( u v ) w An additive identity exists in V There is an element 0 in V such that u 0 = u For every element in V, there is an additive inverse Given a in V, there is a b in V such that a b = 0. We write b = - a Just like for addition of real numbers → → → → → → → → → → → → → → → → → → → → → → → → → V is closed under the operation distributes over Addition of reals distributes over is associative A mutiplicative identity exists 4.VS.3 Saturday, February 06, 2010 11:23 PM Section4VS Page 4 A real vector space is a set V of objects together with two operations and satisfying the following properties: (for all u, v, and w in V and real numbers c and d ) V is closed under the operation If u and v are in V, then u v is in V is commutative u v = v u is associative u ( v w ) = ( u v ) w An additive identity exists in V There is an element 0 in V such that u 0 = u For every element in V, there is an additive inverse Given a in V, there is a b in V such that a b = 0. We write b = - a Just like for addition of real numbers → → → → → → → → → → → → → → → → → → → → → → → → → V is closed under the operation distributes over...
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This note was uploaded on 06/07/2011 for the course MATH 125 taught by Professor Tom during the Fall '07 term at University of Illinois at Urbana–Champaign.

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math125-lecture 4.vs - V is closed under the operation is...

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