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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Linear Independence Example Example . Which of the following collection are linearly independent? (a) ( 1 1 1 ) T , ( 1 1 ) T , ( 1 ) T (b) ( 1 1 ) T , ( 1 ) T (c) ( 1 2 4 ) T , ( 2 1 3 ) T , ( 4 1 1 ) T Chapter 3. Vector Spaces Math1111 Linear Independence Example Example . Which of the following collection are linearly independent? (a) ( 1 1 1 ) T , ( 1 1 ) T , ( 1 ) T (b) ( 1 1 ) T , ( 1 ) T (c) ( 1 2 4 ) T , ( 2 1 3 ) T , ( 4 1 1 ) T Theorem 3.3.1 Let x 1 , x 2 , ··· , x n ∈ R n and write X = ( x 1 x 2 ··· x n ) . Then, x 1 , x 2 , ··· , x n is linearly independent if and only if X is nonsingular. Chapter 3. Vector Spaces Math1111 Vector Spaces Homework 6 Reading Textbook  p.134139 Homework 6 Chapter 3  Section 3 Exercises: Qn. 1, 2, 3, 4, 6 Chapter 3. Vector Spaces Math1111 Linear Independence Proof of Thm 3.3.1 Proof of Theorem. Observe that c 1 x 1 + c 2 x 2 + ··· + c n x n = X c where c = ( c 1 c 2 ··· c n ) T . 1 ◦ (“if” part) Given X = ( x 1 x 2 ··· x n ) is nonsingular. Consider c 1 x 1 + c 2 x 2 + ··· + c n x n = ( * ) where c 1 , c 2 , ··· , c n are scalars. Chapter 3. Vector Spaces Math1111 Linear Independence Proof of Thm 3.3.1 Proof of Theorem. Observe that c 1 x 1 + c 2 x 2 + ··· + c n x n = X c where c = ( c 1 c 2 ··· c n ) T . 1 ◦ (“if” part) Given X = ( x 1 x 2 ··· x n ) is nonsingular. Consider c 1 x 1 + c 2 x 2 + ··· + c n x n = ( * ) where c 1 , c 2 , ··· , c n are scalars. Want to show c 1 = c 2 = ··· = c n = is the only choice for ( * ) to hold. . Chapter 3. Vector Spaces Math1111 Linear Independence Proof of Thm 3.3.1 Proof of Theorem. Observe that c 1 x 1 + c 2 x 2 + ··· + c n x n = X c where c = ( c 1 c 2 ··· c n ) T . 1 ◦ (“if” part) Given X = ( x 1 x 2 ··· x n ) is nonsingular. Consider c 1 x 1 + c 2 x 2 + ··· + c n x n = ( * ) where c 1 , c 2 , ··· , c n are scalars. ( * ) can be rewritten as X c = . Want to show c 1 = c 2 = ··· = c n = is the only choice for ( * ) to hold. i.e. c = . Chapter 3. Vector Spaces Math1111 Linear Independence Proof of Thm 3.3.1 Proof of Theorem. Observe that c 1 x 1 + c 2 x 2 + ··· + c n x n = X c where c = ( c 1 c 2 ··· c n ) T . 1 ◦ (“if” part) Given X = ( x 1 x 2 ··· x n ) is nonsingular. Consider c 1 x 1 + c 2 x 2 + ··· + c n x n = ( * ) where c 1 , c 2 , ··· , c n are scalars. ( * ) can be rewritten as X c = . Since X is nonsingular, i.e. X 1 exists, we have X 1 ( X c ) = X 1 c = . Chapter 3. Vector Spaces Math1111 Linear Independence Proof of Thm 3.3.1 Proof of Theorem. Observe that c 1 x 1 + c 2 x 2 + ··· + c n x n = X c where c = ( c 1 c 2 ··· c n ) T ....
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This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.
 Fall '08
 forgot
 Math, Linear Independence, Vector Space

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