# c3s2 - 3.2 Basic Concepts of Vector Spaces 1 Chapter 3...

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3.2 Basic Concepts of Vector Spaces 1 Chapter 3. Vector Spaces 3.2 Basic Concepts of Vector Spaces Defnition 3.2. Given vectors ~v 1 ,~v 2 ,...,~v k V and scalars r 1 ,r 2 ,...,r k R , k X l =1 r l ~v l = r 1 ~v 1 + r 2 ~v 2 + ··· + r k ~v k is a linear combination of ~v 1 ,~v 2 ,...,~v k with scalar coeﬃcients r 1 ,r 2 ,...,r k . Defnition 3.3. Let X be a subset of vector space V .The span of X is the set of all linear combinations of elements in X and is denoted sp( X ). If V =sp( X ) for some Fnite set X ,then V is fnitely generated . Defnition 3.4. A subset W of a vector space V is a subspace of V if W is itself a vector space. Theorem 3.2. Test For Subspace. A subset W of vector space V is a subspace if and only if (1) ~v, ~w W ~v + ~w W , (2) for all r R and for all ~v W we have r~v W .

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3.2 Basic Concepts of Vector Spaces 2 Example. Page 202 number 4. Defninition 3.5. Let X be a set of vectors from a vector space V .A dependence relation in X is an equation of the form
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c3s2 - 3.2 Basic Concepts of Vector Spaces 1 Chapter 3...

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