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# 121 Class Note for MATH 220 at PSU

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COURSE
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KARMA
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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15
Linear Combination and Span Section 44 0 De nitions from 44 1 Let 1117 1127 7 1 be vectors in a vector space V7 then any vector in the form 11111 12112 any where 117 127 7 an are scalars is called a linear combination of1117 1127 7 1 2 The vectors 1117 1127 7 1 in a vector space V7 are said to span V7 if every vector in V can be written as a linear combination of them That is7 for every 1 6 V7 there are scalars7 117 127 7 a such that va111a2112an11n 3 Let 1117 1127 7 11k be vectors in a vector space V7 the span of117 1127 7 11k is the set of all linear combinations of1117 1127 7 11k That is7 span117 1127 7 11k 1 1 11111 12112 unik where 117 127 7 an are arbitrary scalars 0 Theorem from 44 1 For any collection 1117 1127 7 wk of vectors in a vector space V7 the span of117 1127 7 11k is a subspace of V That is7 span 017 1127 7 11k is a subspace of V Linear Independence Section 45 0 De nitions from 45 1 Let 1117 1127 7 1 be vectors in a vector space V The vectors are said to be linearly dependent if there exists n scalars 017 027 7 077 not all zero such that 01111 0202 cnvn 0 If the vectors are not linearly dependent they are said to be linearly independent 2 Another form of the above de nition Let 1117 1127 7 1 be vectors in a vector space V These vectors are said to be linearly independent if 01111 02112 077177 0 1 implies that that 017 027 7 077 are all zero If equation 1 holds and one or more of the scalars is not zero7 then these vectors are said to be linearly dependent o Theorems from 45 1 Let A be an n x 71 matrix with columns 1117 1127 7 1 The columns of A for a linearly independent set if and only if the only solution to Ax 0 is z 0 2 Let A be an n x 71 matrix Then detA 31 0 if and only if the columns and rows of A are linearly independent Basis and Dimension Section 46 0 De nition from 46 1 A nite set of vectors 017 1127 7 077 is a basis for a vector space V if a 017 1127 7 077 is linearly independent b 017 1127 7 en spans V The dimension of a vector space V is the number of vectors in every basis for V The dimension of a vector space is denoted dim V If every basis has in nitely many vectors7 then V is said to be in nite dimensional7 and if the only vector in its basis is the zero vector then dim V 0 o Theorems from 46 1 117 127 lf117 1127 7 077 is a basis for a vector space V7 then any vector in V can be represented as a unique linear combination of the vectors in the basis 7 U77 and U17 u27 7 um are bases for V then m n This says that the number of terms in all bases for a vector space is the same If H is a subspace of a vector space V7 then dim H S dim V

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