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

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Unformatted text preview: 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. 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 = . 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 = . 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 = . Let c i 6 = 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 = . Let c i 6 = 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 . 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)....
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Lect13 - Chapter 3. Vector Spaces Math1111 Linear...

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