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Math_294Prelim2solnew.08

Math_294Prelim2solnew.08 - Math 294 Prelim 2 solution...

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Math 294 Prelim 2 solution Spring 2008 1. The matrix A of a linear transformation T from 2 3 R R is 1 1 1 0 0 1 A = 1a. Find a basis for Im A , what is the dimension of Im A T ? 1b. Find a basis for Ker A T 1c Is the equation Ax b = G G consistent where (3,3,3) T b = G , if not, find the least square solution of Ax b = G G . Solution : (This problem is a modification of a HW in 5.4, pb20) 1a: the two columns of A are linear independent vectors, so Im A is the span of these two column vectors. Since the rankA T = rankA, dim Im A T =dim Im A = 2. 1b: The Ker A T is the solution of the equation 1 1 0 1 0 1 0 x = G G It is also the vector subspace of 3 R which is orthogonal to Im A . By inspection, the vector ( 1,1,1) T is orthogonal to the columns of A which spans the ImA. Since the KerA T = ( ) Im A and the dim(Im A) = 2, the dim( KerA T ) = 1, so ( ) ( 1,1,1) T T KerA span = . 1c. A simple calculation using RREF shows that the system Ax b = G G is inconsistent. The solution of 1c is a HW problem, pb. 20, section 5.4. 2. Solution : 2a: [ ] 1 1 2 3 1 1 2 1 1 1 2 1 0 2 2 M f f f M     = = + + =       B where 1 2 3 1 0 1 1 0 0 , , 0 0 0 0 0 1 f f f = = =
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