tat02 - THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Summer...

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Unformatted text preview: THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Summer School Math2061 2010 Tutorial 2 (week 2) Solutions 1. For each of the sets of vectors X R 3 below, explicitly describe all of the vectors in the subspace Span( X ) of R 3 . a ) X = { } . b ) X = n 1 1 1 o . c ) X = n 1 1 1 , 2 2 2 o . d ) X = n 1 1 1 , 1 o . e ) X = n 1 1 1 , 1 , 1 1 o . Solution Recall that if X = { v 1 , . . . , v k } then Span( X ) = { 1 v 1 + + k v k | 1 , . . . , k R } . a ) Span( ) = n o . b ) Span 1 1 1 = n r r r r R o . c ) As 2 2 2 = 2 1 1 1 , or by part (b), Span 1 1 1 , 2 2 2 = n r r r r R o . d ) Span 1 1 1 , 1 = n r + s r r r, s R o . We can also describe Span( X ) = n x y z y = z o geometrically as the plane in R 3 with equation y- z = 0 . To see this observe that a vector x y z Span( X ) if and only if x y z = r 1 1 1 + s 1 , for some r, s R . Using Gaussian elimination we can simplify this to 1 1 x 1 y 1 z ! R 2 := R 1- R 2 R 3 := R 1- R 3------- 1 1 x 1 x- y 1 x- z ! R 3 := R 3- R 2------- 1 1 x 1 x- y y- z ! Hence, x y z Span( X ) if and only if y- z = 0 ; that is, if and only if y = z . Conse- quently, we have that Span( X ) = n r + s r r r, s R o = n r s s r, s R o e ) Span 1 1 1 , 1 , 1 1 = n r + s + t r + t r r, s, t R o . Again, we analyze Span( X ) alge- braically. A vector x y z Span( X ) if and only if x y z = a 1 1 1 + b 1 + c 1 1 , Summer School Math2061 Tutorial 2 (week 2) Solutions Page 2 for some a, b, c R . Using Gaussian elimination we can simplify this to 1 1 1 x 1 1 y 1 z ! R 2 := R 1- R 2 R 3 := R 1- R 3------- 1 1 1 x 1 x- y 1 x- z ! Hence, c = x- z , b = x- y and a = x- ( x- y )- ( x- z ) =- x + y + z . In particular, for any vector x y z we can always find suitable a, b, c R , so Span( X ) = n r s t r, s, t R o = R 3 . Note that we now have given two quite different explicit descriptions of Span( X ) : n r + s + t r + t r r, s, t R o = Span( X ) = n r s t r, s, t R o . A priori , it is not obvious that the left and right hand sides of this equation give the same subspace of R 3 . 2. Recall that F is the vector space of functions from R to R , with the usual operations of addition and scalar multiplication of functions. Determine which of the following subsets of F are vector subspaces of F and, in each case, find two functions which are in the set....
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tat02 - THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Summer...

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