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Lect12

# Lect12 - Chapter 3 Vector Spaces Math1111 Span of Vectors...

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Chapter 3. Vector Spaces Math1111 Span of Vectors Example Example . Let e 1 = ( 1 0 0 ) T , e 2 = ( 0 1 0 ) T , v = ( 0 1 2 ) T . 1. Evaluate (i) Span ( v ) , (ii) Span ( e 1 , v ) , (iii) Span ( e 1 , e 2 , v ) . 2. Describe them geometrically. Ans . (i) Span ( v ) = { ( 0 α 2 α ) : α R } (ii) Span ( e 1 , e 2 ) = { ( α β 0 ) T : α , β R } (iii) Span ( e 1 , e 2 , v ) = R 3 . Theorem 3.2.1 Let v 1 , v 2 , ··· , v n V (vector space). Then, Span ( v 1 , v 2 , ··· , v n ) is a subspace of V .

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Chapter 3. Vector Spaces Math1111 Span of Vectors Spanning set - Definition Definition If every vector in V is a linear combination of v 1 , v 2 , ··· , v n , in other words, V = Span ( v 1 , v 2 , ··· , v n ) , then we call the set { v 1 , v 2 , ··· , v n } a spanning set for V .
Chapter 3. Vector Spaces Math1111 Span of Vectors Spanning set - Definition Definition If every vector in V is a linear combination of v 1 , v 2 , ··· , v n , in other words, V = Span ( v 1 , v 2 , ··· , v n ) , then we call the set { v 1 , v 2 , ··· , v n } a spanning set for V . Example . Which of the following are spanning sets for R 3 ? (a) { e 1 , e 2 , e 3 , ( 1 2 3 ) T } (b) { ( 1 1 1 ) T , ( 1 1 0 ) T , ( 1 0 0 ) T } (c) { ( 1 0 1 ) T , ( 0 1 0 ) T } (d) { ( 1 2 4 ) T , ( 2 1 3 ) T , ( 4 - 1 1 ) T }

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Chapter 3. Vector Spaces Math1111 Span of Vectors Spanning set - Definition Definition If every vector in V is a linear combination of v 1 , v 2 , ··· , v n , in other words, V = Span ( v 1 , v 2 , ··· , v n ) , then we call the set { v 1 , v 2 , ··· , v n } a spanning set for V . Example . Which of the following are spanning sets for R 3 ? (a) { e 1 , e 2 , e 3 , ( 1 2 3 ) T } (b) { ( 1 1 1 ) T , ( 1 1 0 ) T , ( 1 0 0 ) T } (c) { ( 1 0 1 ) T , ( 0 1 0 ) T } (d) { ( 1 2 4 ) T , ( 2 1 3 ) T , ( 4 - 1 1 ) T } Note - How to check spanning sets? By Thm 3.2.1, suffices to check: every vector in V belongs to the span.
Chapter 3. Vector Spaces Math1111 Span of Vectors Spanning set - Example Example . Let P 3 be the vector space consisting of all polynomials of degree < 3 . Let v 1 = 1 - x 2 , v 2 = x + 2 , v 3 = x 2 . Show that { v 1 , v 2 , v 3 } is a spanning set for P 3 .

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Chapter 3. Vector Spaces Math1111 Span of Vectors Spanning set - Example Example . Let P 3 be the vector space consisting of all polynomials of degree < 3 . Let v 1 = 1 - x 2 , v 2 = x + 2 , v 3 = x 2 . Show that { v 1 , v 2 , v 3 } is a spanning set for P 3 . Proof. Any element in P 3 is of the form p ( x ) = ax 2 + bx + c .
Chapter 3. Vector Spaces Math1111 Span of Vectors Spanning set - Example Example . Let P 3 be the vector space consisting of all polynomials of degree < 3 . Let v 1 = 1 - x 2 , v 2 = x + 2 , v 3 = x 2 . Show that { v 1 , v 2 , v 3 } is a spanning set for P 3 . Proof. For any p ( x ) = ax 2 + bx + c , want to find α , β , γ R such that p ( x ) = α v 1 + β v 2 + γ v 3 . Any element in P 3 is of the form p ( x ) = ax 2 + bx + c .

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Chapter 3. Vector Spaces Math1111 Span of Vectors Spanning set - Example Example . Let P 3 be the vector space consisting of all polynomials of degree < 3 . Let v 1 = 1 - x 2 , v 2 = x + 2 , v 3 = x 2 . Show that { v 1 , v 2 , v 3 } is a spanning set for P 3 . Proof. For any p ( x ) = ax 2 + bx + c , want to find α , β , γ R such that p ( x ) = α v 1 + β v 2 + γ v 3 . As α v 1 + β v 2 + γ v 3 = α ( 1 - x 2 ) + β ( x + 2 ) + γ x 2 = ( α + 2 β ) + β x + ( γ - α ) x 2 , it suffices to solve the system α + 2 β = c , β = b , - α + γ = a .
Chapter 3. Vector Spaces Math1111 Vector Spaces Homework 5 Reading Leon (7th edition): p.128 - p.131 Leon (8th edition): p.121 - p.125 Homework 5 Leon (7th edition): Chapter 3 - Section 2 Qn. 9-14 .

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