Lect16 - Chapter 3 Vector Spaces Math1111 Basis and...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Basis and Dimension Theorem 3.4.3 Theorem 3.4.3 Let V be a vector space and dim V = n ≥ 1 . I) Any set of n linearly independent vectors spans V . II) Any n vectors that span V are linearly independent. Chapter 3. Vector Spaces Math1111 Basis and Dimension Theorem 3.4.3 Theorem 3.4.3 Let V be a vector space and dim V = n ≥ 1 . I) Any set of n linearly independent vectors spans V . Proof. Let v 1 , ··· , v n be linearly independent vectors in V . Suppose V 6 = Span ( v 1 , ··· , v n ) . Take v ∈ V but v / ∈ Span ( v 1 , ··· , v n ) . Recall Span ( v 1 , ··· , v n ) ⊂ V Chapter 3. Vector Spaces Math1111 Basis and Dimension Theorem 3.4.3 Theorem 3.4.3 Let V be a vector space and dim V = n ≥ 1 . I) Any set of n linearly independent vectors spans V . Proof. Let v 1 , ··· , v n be linearly independent vectors in V . Suppose V 6 = Span ( v 1 , ··· , v n ) . Take v ∈ V but v / ∈ Span ( v 1 , ··· , v n ) . By Thm 3.4.1, v 1 , ··· , v n , v are linearly dependent. Because dim V = n but v 1 , ··· , v n , v are n + 1 vectors Chapter 3. Vector Spaces Math1111 Basis and Dimension Theorem 3.4.3 Theorem 3.4.3 Let V be a vector space and dim V = n ≥ 1 . I) Any set of n linearly independent vectors spans V . Proof. Let v 1 , ··· , v n be linearly independent vectors in V . Suppose V 6 = Span ( v 1 , ··· , v n ) . Take v ∈ V but v / ∈ Span ( v 1 , ··· , v n ) . By Thm 3.4.1, v 1 , ··· , v n , v are linearly dependent. ∴ There exist scalars c 1 , ··· , c n , c n + 1 , not all zero, such that c 1 v 1 + ··· + c n v n + c n + 1 v = Claim: c n + 1 6 = . Chapter 3. Vector Spaces Math1111 Basis and Dimension Theorem 3.4.3 Theorem 3.4.3 Let V be a vector space and dim V = n ≥ 1 . I) Any set of n linearly independent vectors spans V . Proof. Let v 1 , ··· , v n be linearly independent vectors in V . Suppose V 6 = Span ( v 1 , ··· , v n ) . Take v ∈ V but v / ∈ Span ( v 1 , ··· , v n ) . By Thm 3.4.1, v 1 , ··· , v n , v are linearly dependent. ∴ There exist scalars c 1 , ··· , c n , c n + 1 , not all zero, such that c 1 v 1 + ··· + c n v n + c n + 1 v = Claim: c n + 1 6 = . Think about what happen if c n + 1 = Chapter 3. Vector Spaces Math1111 Basis and Dimension Theorem 3.4.3 Theorem 3.4.3 Let V be a vector space and dim V = n ≥ 1 . I) Any set of n linearly independent vectors spans V . Proof. Let v 1 , ··· , v n be linearly independent vectors in V . Suppose V 6 = Span ( v 1 , ··· , v n ) . Take v ∈ V but v / ∈ Span ( v 1 , ··· , v n ) . By Thm 3.4.1, v 1 , ··· , v n , v are linearly dependent. ∴ There exist scalars c 1 , ··· , c n , c n + 1 , not all zero, such that c 1 v 1 + ··· + c n v n + c n + 1 v = Claim: c n + 1 6 = . ∴ v =- c 1 c n + 1 v 1 + ··· +- c n c n + 1 v n Chapter 3. Vector Spaces Math1111 Basis and Dimension Theorem 3.4.3 Theorem 3.4.3 Let V be a vector space and dim V = n ≥ 1 ....
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Lect16 - Chapter 3 Vector Spaces Math1111 Basis and...

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