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Lect13

# Lect13 - Chapter 3 Vector Spaces Math1111 Linear...

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Chapter 3. Vector Spaces Math1111 Linear Independence Observation Through the last example, we have the following observations. Theorem Let V be a vector space and let v 1 , v 2 , ··· , v n V . I. Suppose v 1 , v 2 , ··· , v n span V . If v i is a linear combination of the other n - 1 vectors, then { v 1 , ··· , v i - 1 , v i + 1 , ··· , v n } is a spanning set for V .
Chapter 3. Vector Spaces Math1111 Linear Independence Observation Through the last example, we have the following observations. Theorem Let V be a vector space and let v 1 , v 2 , ··· , v n V . I. Suppose v 1 , v 2 , ··· , v n span V . If v i is a linear combination of the other n - 1 vectors, then { v 1 , ··· , v i - 1 , v i + 1 , ··· , v n } is a spanning set for V . Proof of (I). Straightforward.

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Chapter 3. Vector Spaces Math1111 Linear Independence Observation Through the last example, we have the following observations. Theorem Let V be a vector space and let v 1 , v 2 , ··· , v n V . I. Suppose v 1 , v 2 , ··· , v n span V . If v i is a linear combination of the other n - 1 vectors, then { v 1 , ··· , v i - 1 , v i + 1 , ··· , v n } is a spanning set for V . II. One of v 1 , ··· , v n is a linear combination of the other n - 1 vectors if and only if there exist scalars c 1 , c 2 , ··· , c n not all zero such that c 1 v 1 + c 2 v 2 + ··· + c n v n = 0 .
Chapter 3. Vector Spaces Math1111 Linear Independence Observation (Cont’d) Proof of (II). "If" part: Suppose there exist scalars c 1 , c 2 , ··· , c n not all zero such that c 1 v 1 + c 2 v 2 + ··· + c n v n = 0 .

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Chapter 3. Vector Spaces Math1111 Linear Independence Observation (Cont’d) Proof of (II). "If" part: Suppose there exist scalars c 1 , c 2 , ··· , c n not all zero such that c 1 v 1 + c 2 v 2 + ··· + c n v n = 0 . Let c i 6 = 0 where 1 i n . Then, - c i v i = c 1 v 1 + ··· + c i - 1 v i - 1 + c i + 1 v i + 1 + ··· + c n v n v i = - c 1 c i v 1 + ··· + - c n c i v n .
Chapter 3. Vector Spaces Math1111 Linear Independence Observation (Cont’d) Proof of (II). "If" part: Suppose there exist scalars c 1 , c 2 , ··· , c n not all zero such that c 1 v 1 + c 2 v 2 + ··· + c n v n = 0 . Let c i 6 = 0 where 1 i n . Then, - c i v i = c 1 v 1 + ··· + c i - 1 v i - 1 + c i + 1 v i + 1 + ··· + c n v n v i = - c 1 c i v 1 + ··· + - c n c i v n . v i is a linear combination of the other n - 1 vectors v 1 , ··· , v i - 1 , v i + 1 , ··· , v n .

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Chapter 3. Vector Spaces Math1111 Linear Independence Observation (Cont’d) Proof of (II). "Only if" part: Suppose one of v 1 , ··· , v n is a linear combination of the other n - 1 vectors.
Chapter 3. Vector Spaces Math1111 Linear Independence Observation (Cont’d) Proof of (II). "Only if" part: Suppose one of v 1 , ··· , v n is a linear combination of the other n - 1 vectors. W.L.O.G. ( = Without loss of generality), let v 1 be a linear combination of the other n - 1 vectors.

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Chapter 3. Vector Spaces Math1111 Linear Independence Observation (Cont’d) Proof of (II). "Only if" part: Suppose one of v 1 , ··· , v n is a linear combination of the other n - 1 vectors.
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Lect13 - Chapter 3 Vector Spaces Math1111 Linear...

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