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lecture10 - Lecture Lecture 10 3.2 Linear Combinations and...

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Lecture 10 3.2 Linear Combinations and Linear Spans
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Section 3 2 Section 3.2 Linear Combinations and Linear Spans Objective • What is a linear span ? • What is a subspace ? • What are some subspaces of R n ? •What is a solution space of a LS? What is a • How to show a linear span is contained in another? Other terminologies t d d t” t Chapter 3 Vector Spaces 2 vector space, zero space, “redundant” vectors closure property
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Int od ction Introduction (-1, 1, 1), (7, 2, -1) and (0, 0, 0) These vectors are called They are all of the form: s (2, 1, 0) + t (-3, 0, 1) linear combinations of (2,1,0) and (-3,0,1) How many linear combinations of (2,1,0) and (-3,0,1) are there? Infinite The set of all linear combinations of (2,1,0) d ( 3 0 1) and (-3,0,1) {s (2, 1, 0) + t (-3, 0, 1 ) | s, t œ R } We call it: the linear span of (2 1 0) and ( 3 0 1) Chapter 3 Vector spaces 3 (2,1,0) (-3,0,1) We write it: span{ (2,1,0) , (-3,0,1) }
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What is a linear span? Definition 3.2.3.1 S = { u 1 , u 2 , …, u k } : a (finite) subset of R n . The set of all linear combinations of u 1 , u 2 , …, u k { c 1 u 1 + c 2 u 2 + ··· + c k u k | c 1 , c 2 , …, c k in R } This set is called 1 1 + 2 2 + ··· + c 1 , c 2 , …, in R = = the linear span of u 1 , u 2 , …, u k It is an (infinite) subset of R n . or the linear span of S Linear span ” is always used with a set of vectors Chapter 3 Vector spaces 4 This set is denoted by span{ u 1 , u 2 , …, u k } or span( S )
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Express a linear span in explicit set notation form Example 3.2.6.1 S = {(1, 0, 0, 1), (0, 1, 1, 0)} Œ R 4 A general vector in span( S ): a (1, 0, 0, –1) + b (0, 1, 1, 0) = ( a , b , b , – a ). = { ( a b b a ) | a b œ R } explicit form span( S ) = { a (1, 0, 0, –1) + b (0, 1, 1, 0)| a , b œ R } , , , , = span{(1, 0, 0, 1), (0, 1, 1, 0)} linear span form Not every element of R 4 is an element of span( S ). span( S ) Œ R 4 . Chapter 3 Vector spaces 5 So span( S ) R 4 .
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What is a linear span? Definition 3.2.3.1 S = { u 1 , u 2 , …, u k } : a (finite) subset of R n . R n Span( S ) c 1 u 1 + c 2 u 2 + ··· + c k u k S Œ R n S Œ span( S ) u 1 + 2 u 2 + ··· + 7 u k - 3 u 1 + 0 u 2 + ··· + 2 u k span( S ) Œ R n u 1 , u 2 , …, u k S Chapter 3 Vector spaces 6
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Geometrical meaning of linear span Discussion 3.2.7 span = extend across (Oxford Dictionary) S = { u } span(S) = span{ u } = { c u | c in R } u the origin In R 2 and R 3 the line through the origin and parallel to u span(S) = span{ u , v } S = { u , v } 2 3 = { s u + t v | s , t œ R } v u In R and R the plane containing Chapter 3 Vector spaces 7 the origin and parallel to u and v
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How to express a given subset as linear span? Example 3.2.6.3 V 2 = { ( x , y , z ) | x + y z = 0 } subset of R 3 implicit form explicit form V 2 = { ( t - s , s, t ) | s, t œ R } ( t s s t ) = ( t 0 t ) + ( s s 0) t - s , s, , 0, - s , s, 0) = t (1, 0, 1) + s ( - 1, 1, 0) V is the set of all linear combinations of 2 (1, 0, 1) and (-1, 1, 0) V 2 = span{(1 0 1) (-1 1 0)} linear span form ( V 2 is a plane in R 3 .) = span{(1, 0, 1), ( 1, 1, 0)} N t b t f R 4 b d i li Chapter 3 Vector spaces 8 Not every subset of can be expressed in linear span form
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When we span a set S, we get the span of S.
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