HM9-soln - Solution for HW9 December 6 2011 4.115 Since dim U,dim W ≤ dim U W ≤ dim V we can get dim U W can be 5,6 and 7 And then from dim U

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Unformatted text preview: Solution for HW9 December 6, 2011 4.115 Since dim ( U ) ,dim ( W ) ≤ dim ( U + W ) ≤ dim ( V ) , we can get dim ( U + W ) can be 5,6 and 7. And then from dim ( U ∩ W ) = dim ( U ) + dim ( W )- dim ( U + W ) we can see dim ( U ∩ W ) can be 4,3 and 2. 4.117 (a) A vector v = ( x,y,z,s,t ) ∈ U if and only if v is a linear combination of the vectors in U . We can put u 1 ,u 2 ,u 3 ,v as columns of a matrix and reduce the matrix to echelon form, which yields 1 1 1 x- 1- 2- 1 y- 1- 2- 2 z- 2- 2 s- 3 1 t ∼ 1 1 1 x- 1 x + y- 1- 1 x + z 2 2 x + s- 3 1 t ∼ 1 1 1 x- 1 x + y- 1- y + z 4 x + 2 y + s 1- 3 x- 3 y + t ∼ 1 1 1 x- 1 x + y- 1- y + z 4 x + 2 y + s- 3 x- 4 y + z + t Here we can get the required homogeneous system 4 x + 2 y + s =- 3 x- 4 y + z + t = Using the same technique we can again reduce the matrix with w 1 ,w 2 ,w 3 ,v as columns to echelon form 1 1 1 x- 2- 1- 1 y- 3- 3- 2 z 2 2 s- 2- 4- 5 t ∼ 1 1 1 x 1 1 2 x + y 1 3 x + z 2 2 s- 2- 3 2 x + t ∼ 1 1 1 x 1 1 2 x + y 1 3 x + z- 4 x- 2 y + s 9 x + 2 y + z + t 1 Here we can get the required homogeneous system- 4 x- 2 y + s = 9 x + 2 y + z + t = (b) Combine both of the above systems to obtain a homogeneous system, whose solution space is U ∩ W , and reduce the system to echelon form, yielding 4 2 1- 3- 4 1 1- 4- 2 1 9 2 1 1 ∼ 1- 2 1 1 1- 3- 4 1 1 2 9 2 1 1 ∼ 1- 2 1 1- 10 4 4 2 20- 8- 8 ∼ 1- 2 1 1- 5 2 2 2 There are two free variables z,t . And set z = 5 ,t = 0 and z = 0 ,t = 5 will give the basis of U ∩ W , { (- 1 , 2 , 5 , , 0) , (- 1 , 2 , , , 5) } . 4.130 Let v = x ( t 3 + t 2 ) + y ( t 2 + t ) + z ( t + 1) + s = xt 3 + ( x + y ) t 2 + ( y + z ) t + ( z + s ) . (a) v = 2 t 3 + t 2- 4 t + 2 gives us x = 2 x + y = 1 y + z =- 4 z + s = 2 and we can get the solution (2 ,- 1 ,- 3 , 5) , which is the coordinate vector we want. (b) v = at 3 + bt 2 + ct + d gives us x = a x + y = b y + z = c z + s = d and we can get the solution ( a,- a + b,a- b + c,- a + b- c + d ) , which is the coordinate vector we want....
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This note was uploaded on 02/28/2012 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.

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HM9-soln - Solution for HW9 December 6 2011 4.115 Since dim U,dim W ≤ dim U W ≤ dim V we can get dim U W can be 5,6 and 7 And then from dim U

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