Lect15 - Chapter 3 Vector Spaces Math1111 Linear...

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Unformatted text preview: 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 . 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 = . Hence ( * ) holds valid only for c 1 = c 2 = ··· = c n = . i.e. x 1 , ··· , x n are linearly independent. 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 . 2 ◦ (“only if” part) Suppose x 1 , x 2 , ··· , x n is linear independent. 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 04/30/2010 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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Lect15 - Chapter 3 Vector Spaces Math1111 Linear...

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