Lect10

# Lect10 - Chapter 3. Vector Spaces Math1111 Vector Spaces...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . Let W = ( a 1 ) T : a ∈ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T ∈ W , (ii) α ( a 1 ) T = ( α a 1 ) T for any scalar α and ( a 1 ) T ∈ W . Is W a vector space with respect to these two operations? Yes Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . Let W = ( a 1 ) T : a ∈ R . Define (i) ( x 1 ) T + ( y 1 ) T = ( x + y 1 ) T for any ( x 1 ) T , ( y 1 ) T ∈ W , (ii) α ( a 1 ) T = ( α a 1 ) T for any scalar α and ( a 1 ) T ∈ W . Is W a vector space with respect to these two operations? Yes Theorem 3.1.1 Let V be a vector space and x ∈ V . Then (i) x = , (ii) x + y = implies y =- x (i.e. additive inverse is unique) , (ii)’ Zero element is unique, (iii) (- 1 ) x =- x . (i.e. The scalar multiple (- 1 ) x is the additive inverse of x .) Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . The set R n with the usual addition and scalar multiplication is a vector space. Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . The set R n with the usual addition and scalar multiplication is a vector space. Example . Let P n be the set of all polynomials of degree ≤ n . Under the usual operations, P n is a vector space. ( Addition: Let p , q ∈ P n . Define p + q as ( p + q ) ( x ) = p ( x ) + q ( x ) . Scalar multiplication: Let α ∈ R & p ∈ P n . Define α p as ( α p ) ( x ) = α p ( x ) . ) Chapter 3. Vector Spaces Math1111 Vector Spaces Example & Properties Example . The set R n with the usual addition and scalar multiplication is a vector space. Example . Let P n be the set of all polynomials of degree ≤ n . Under the usual operations, P n is a vector space. ( Addition: Let p , q ∈ P n . Define p + q as ( p + q ) ( x ) = p ( x ) + q ( x ) . Scalar multiplication: Let α ∈ R & p ∈ P n . Define α p as ( α p ) ( x ) = α p ( x ) . ) Example . Let C [ a , b ] be the set of all real-valued functions on the interval [ a , b ] . Under the usual operations, C [ a , b ] is a vector space. Chapter 3. Vector Spaces Math1111 Subspaces Exercise Exercise Let V = { f : f is differentiable infinitely many times on...
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Lect10 - Chapter 3. Vector Spaces Math1111 Vector Spaces...

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