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Lect14

# Lect14 - Chapter 3 Vector Spaces Math1111 Linear...

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Chapter 3. Vector Spaces Math1111 Linear Independence Example Example . Which of the following collection are linearly independent? (a) ( 1 1 1 ) T , ( 1 1 0 ) T , ( 1 0 0 ) T (b) ( 1 0 1 ) T , ( 0 1 0 ) T (c) ( 1 2 4 ) T , ( 2 1 3 ) T , ( 4 - 1 1 ) T

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Chapter 3. Vector Spaces Math1111 Linear Independence Example Example . Which of the following collection are linearly independent? (a) ( 1 1 1 ) T , ( 1 1 0 ) T , ( 1 0 0 ) T (b) ( 1 0 1 ) T , ( 0 1 0 ) 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 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.134-139 Homework 6 Chapter 3 - Section 3 Exercises: Qn. 1, 2, 3, 4, 6

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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 = 0 ( * ) 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 = 0 ( * ) where c 1 , c 2 , ··· , c n are scalars. Want to show c 1 = c 2 = ··· = c n = 0 is the only choice for ( * ) to hold. .

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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 = 0 ( * ) where c 1 , c 2 , ··· , c n are scalars. ( * ) can be rewritten as X c = 0 . Want to show c 1 = c 2 = ··· = c n = 0 is the only choice for ( * ) to hold. i.e. c = 0 .
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 = 0 ( * ) where c 1 , c 2 , ··· , c n are scalars. ( * ) can be rewritten as X c = 0 . Since X is nonsingular, i.e. X - 1 exists, we have X - 1 ( X c ) = X - 1 0 c = 0 .

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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 = 0 ( * ) where c 1 , c 2 , ··· , c n are scalars. ( * ) can be rewritten as X c = 0 . Since X is nonsingular, i.e. X - 1 exists, we have X - 1 ( X c ) = X - 1 0 c = 0 . Hence ( * ) holds valid only for c 1 = c 2 = ··· = c n = 0 . 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 .

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Lect14 - Chapter 3 Vector Spaces Math1111 Linear...

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