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| let V be a set of objects in which two operations (addition and scalar multiplication) are defined. if the listed properties (aka axioms) are satisfied for every u,v,and w in V and every scalar (real number c and d) then V is called a vector space |
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| V has a zero vector such that for every u in V |
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| for every u in V, there is a vector in V denoted by -u |
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such that u+(-u)=0
additive inverses property |
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| closure under scalar multiplication |
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| associative property of multiplication |
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| four things needed in order to have a vector space |
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1. set of vectors
2. set of scalars
3. 2 operations
even though the objects in vector space v are called vectors, the vectors in v could be matrices, or polynomials, or anything else that satisfies the axiom |
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| set of all ordered triples |
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| set of all continuous funtions defined (-inf., inf.) |
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| set of all continuous functions defined on [a,b] |
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| set of all polynomials (anxn+an-1xn-1+...a1x1+a0) |
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| set of all polynomials of degree equal or greater to n |
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| properties of scalar multiplication |
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let v be an element of a vector space V1 and let c be any scalar then the following are true
1. 0*V=0
2. c*0=0
3. if cV=0, then c=0 or V=0
4. (-1)V=-V |
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