math125-lecture 4.4 - Solution set-2 t s 1 t s | t and s...

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Unformatted text preview: Solution set: { ( -2 t + s + 1, t , s ) | t and s are in R } 2 parameters → a plane Sec 4.4 Saturday, February 06, 2010 11:23 PM Section4dot4 Page 1 Solution set: { ( -2 t + s + 1, t , s ) | t and s are in R } 2 parameters → a plane x = ( x, y, z ) = ( -2 t + s + 1, t , s ) = ( 1, 0, 0 ) + ( -2 t , t , 0 ) + ( s , 0, s ) = ( 1, 0, 0 ) + t ( -2, 1, 0 ) + s ( 1, 0, 1 ) = ( 1, 0, 0 ) + span{ ( -2, 1, 0 ), ( 1, 0, 1 ) } → 4.4.1 Saturday, February 06, 2010 11:23 PM Section4dot4 Page 2 Solution set: { ( -2 t + s + 1, t , s ) | t and s are in R } 2 parameters → a plane x = ( x, y, z ) = ( -2 t + s + 1, t , s ) = ( 1, 0, 0 ) + ( -2 t , t , 0 ) + ( s , 0, s ) = ( 1, 0, 0 ) + t ( -2, 1, 0 ) + s ( 1, 0, 1 ) = ( 1, 0, 0 ) + span{ ( -2, 1, 0 ), ( 1, 0, 1 ) } → The vector equation of a plane in R n is given by x = P + t v 1 + s v 2 , t and s in R where P is any point in the plane and v 1 and v 2 are any two vectors that are parallel to the plane but NOT parallel to each other → → → → → → → 4.4.2 Saturday, February 06, 2010 11:23 PM Section4dot4 Page 3 Solution set: { ( -2 t + s + 1, t , s ) | t and s are in R } 2 parameters → a plane x = ( x, y, z ) = ( -2 t + s + 1, t , s ) = ( 1, 0, 0 ) + ( -2 t , t , 0 ) + ( s , 0, s ) = ( 1, 0, 0 ) + t ( -2, 1, 0 ) + s ( 1, 0, 1 ) = ( 1, 0, 0 ) + span{ ( -2, 1, 0 ), ( 1, 0, 1 ) } → The vector equation of a plane in R n is given by x = P + t v 1 + s v 2 , t and s in R where P is any point in the plane and v 1 and v 2 are any two vectors that are parallel to the plane but NOT parallel to each other When P = 0 , the plane intersects the origin and so can be written as span{ v 1 , v 2 }. In this case the plane is a subspace of R n → → → → → → → → → → → 4.4.3 (*) Saturday, February 06, 2010 11:23 PM Section4dot4 Page 4 A span of a single (non-zero) vector is a line. Is the span of 2 vectors a plane? Is the span of 3 vectors a hyperplane? Is 4.4.4 Monday, February 15, 2010 10:28 PM Section4dot4 Page 5 A span of a single (non-zero) vector is a line. Is the span of 2 vectors a plane? Is the span of 3 vectors a hyperplane? NOT ALWAYS ! It depends on how the spanning vectors are related to each other Is 4.4.5 Monday, February 15, 2010 10:28 PM Section4dot4 Page 6 A span of a single (non-zero) vector is a line. Is the span of 2 vectors a plane? Is the span of 3 vectors a hyperplane? NOT ALWAYS ! It depends on how the spanning vectors are related to each other Let S = { v 1 , v 2 , . . . , v k } . We say that S is a) linearly dependent if there exist constants c 1 , c 2 , . . . , c k , NOT ALL ZERO, such that c 1 v 1 + c 2 v 2 + . . . + c k v k = ( at least one vector is a linear combo of the others - why? ) → → → → → → → Is 4.4.6 Monday, February 15, 2010 10:28 PM Section4dot4 Page 7 A span of a single (non-zero) vector is a line....
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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 4.4 - Solution set-2 t s 1 t s | t and s...

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