math125-lecture 4.2 - Let v 1 v 2 v k be vectors in R n A...

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Unformatted text preview: Let v 1 , v 2 , . . ., v k be vectors in R n . A vector v is called a linear combination of v 1 , v 2 , . . ., v k if v = c 1 v 1 + c 2 v 2 + . . . + c k v k for some scalars (i.e. real numbers) c 1 , c 2 , . . ., c k → → → → → → → → → → → Sec 4.2 Saturday, February 06, 2010 11:14 PM Section4dot2 Page 1 Let v 1 , v 2 , . . ., v k be vectors in R n . A vector v is called a linear combination of v 1 , v 2 , . . ., v k if v = c 1 v 1 + c 2 v 2 + . . . + c k v k for some scalars (i.e. real numbers) c 1 , c 2 , . . ., c k → → → → → → → → → → → We want to know if there are real numbers c 1 , c 2 , and c 3 such that v = c 1 u 1 + c 2 u 2 + c 3 u 3 i.e. ( 2, 1, 5 ) = c 1 ( 1, 2, 1 ) + c 2 ( 1, 0, 2 ) + c 3 ( 1, 1, 0 ) → → → → 4.2.1 Saturday, February 06, 2010 11:14 PM Section4dot2 Page 2 Let v 1 , v 2 , . . ., v k be vectors in R n . A vector v is called a linear combination of v 1 , v 2 , . . ., v k if v = c 1 v 1 + c 2 v 2 + . . . + c k v k for some scalars (i.e. real numbers) c 1 , c 2 , . . ., c k → → → → → → → → → → → We want to know if there are real numbers c 1 , c 2 , and c 3 such that v = c 1 u 1 + c 2 u 2 + c 3 u 3 i.e. ( 2, 1, 5 ) = c 1 ( 1, 2, 1 ) + c 2 ( 1, 0, 2 ) + c 3 ( 1, 1, 0 ) i.e. ( 2, 1, 5 ) = ( c 1 + c 2 + c 3 , 2 c 1 + 0 c 2 + c 3 , c 1 + 2 c 2 + 0 c 3 ) → → → → 4.2.2 Saturday, February 06, 2010 11:14 PM Section4dot2 Page 3 Let v 1 , v 2 , . . ., v k be vectors in R n . A vector v is called a linear combination of v 1 , v 2 , . . ., v k if v = c 1 v 1 + c 2 v 2 + . . . + c k v k for some scalars (i.e. real numbers) c 1 , c 2 , . . ., c k → → → → → → → → → → → We want to know if there are real numbers c 1 , c 2 , and c 3 such that v = c 1 u 1 + c 2 u 2 + c 3 u 3 i.e. ( 2, 1, 5 ) = c 1 ( 1, 2, 1 ) + c 2 ( 1, 0, 2 ) + c 3 ( 1, 1, 0 ) i.e. ( 2, 1, 5 ) = ( c 1 + c 2 + c 3 , 2 c 1 + 0 c 2 + c 3 , c 1 + 2 c 2 + 0 c 3 ) As a linear system: c 1 + c 2 + c 3 = 2 2 c 1 + c 3 = 1 c 1 + 2 c 2 = 5 → → → → 4.2.3 Saturday, February 06, 2010 11:14 PM Section4dot2 Page 4 Let v 1 , v 2 , . . ., v k be vectors in R n . A vector v is called a linear combination of v 1 , v 2 , . . ., v k if v = c 1 v 1 + c 2 v 2 + . . . + c k v k for some scalars (i.e. real numbers) c 1 , c 2 , . . ., c k → → → → → → → → → → → We want to know if there are real numbers c 1 , c 2 , and c 3 such that v = c 1 u 1 + c 2 u 2 + c 3 u 3 i.e. ( 2, 1, 5 ) = c 1 ( 1, 2, 1 ) + c 2 ( 1, 0, 2 ) + c 3 ( 1, 1, 0 ) i.e. ( 2, 1, 5 ) = ( c 1 + c 2 + c 3 , 2 c 1 + 0 c 2 + c 3 , c 1 + 2 c 2 + 0 c 3 ) As a linear system: c 1 + c 2 + c 3 = 2 2 c 1 + c 3 = 1 c 1 + 2 c 2 = 5 As an augmented matrix: 1 1 1 2 2 0 1 1 1 2 0 5 → → → → 4.2.4 Saturday, February 06, 2010 11:14 PM Section4dot2 Page 5 Let v 1 , v 2 , . . ., v k be vectors in R n . A vector v is called a linear combination of v 1 , v 2 , . . ., v k if v = c 1 v 1 + c 2 v 2 + . . . + c k v k for some scalars (i.e. real numbers) for some scalars (i....
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This note was uploaded on 06/07/2011 for the course MATH 125 taught by Professor Tom during the Fall '07 term at University of Illinois at Urbana–Champaign.

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math125-lecture 4.2 - Let v 1 v 2 v k be vectors in R n A...

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