c3s1 - ~v , (3) if ~u + ~v = ~u + ~w then ~v = ~w , (4) ~v...

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3.1 Vector Spaces 1 Chapter 3. Vector Spaces 3.1 Vector Spaces Defnition 3.1. A vector space is a set V of vectors along with an operation of addition + of vectors and multiplication of a vector by a scalar (real number), which satisFes the following. ±or all ~u,~v, ~w V and for all r, s R : (A1) ( ~u + ~v )+ ~w = ~u +( ~v + ~w ) (A2) ~v + ~w = ~w + ~v (A3) There exists ~ 0 V such that ~ 0+ ~v = ~v (A4) ~v +( ~v )= ~ 0 (S1) r ( ~v + ~w )= r~v + r~w (S2) ( r + s ) ~v = r~v + s~v (S3) r ( s~v )=( rs ) ~v (S4) 1 ~v = ~v Defnition. ~ 0isthe additive identity . ~v is the additive inverse of ~v .
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3.1 Vector Spaces 2 Example. Some examples of vector spaces are: (1) The set of all polynomials of degree n or less, denoted P n . (2) All m × n matrices. (3) The set of all functions integrable f with domain [0 , 1] such that Z 1 0 | f ( x ) | 2 dx < . This vector space is denoted L 2 [0 , 1]: L 2 [0 , 1] = ± f ² ² ² ² Z 1 0 | f ( x ) | 2 dx < ³ . Theorem 3.1. Elementary Properties of Vector Spaces. Every vector space V satisFes: (1) the vector ~ 0 is the unique additive identity in a vector space, (2) for each ~v V , ~v is the unique additive inverse of
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Unformatted text preview: ~v , (3) if ~u + ~v = ~u + ~w then ~v = ~w , (4) ~v = ~ 0 for all ~v V , (5) r ~ 0 = ~ 0 for all scalars r R , (6) ( r ) ~v = r ( ~v ) = ( r~v ) for all r R and for all ~v V . 3.1 Vector Spaces 3 Proof of (1) and (3). Suppose that there are two additive identities, ~ 0 and ~ . Then consider: ~ 0 = ~ 0 + ~ (since ~ is an additive identity) = ~ (since ~ 0 is an additive identity). Therefore, ~ 0 = ~ and the additive identity is unique. Suppose ~u + ~v = ~u + ~w . Then we add ~u to both sides of the equation and we get: ~u + ~v + ( ~u ) = ~u + ~w + ( ~u ) ~v + ( ~u ~u ) = ~w + ( ~u ~u ) ~v + ~ 0 = ~w + ~ ~v = ~w The conclusion holds. QED Example. Page 189 number 14 and page 190 number 24....
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c3s1 - ~v , (3) if ~u + ~v = ~u + ~w then ~v = ~w , (4) ~v...

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