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Evaluate T AB C− given that 2 2 0 1 2 3 3 1 , and 0 1 3 0 1 2 12 7 A B C − − − = = =
Given the system of equations 2 3 2 11 3 2 3 7 4 4 14 x y z x y z x y z + − = − − + = − + = , find the values of x, y, and z using matrix inversion.
if M (x,y) = (x+y, x+y )
Find the matrix representation of M with respect to (v1,v2) where v1= (1,1) v2 = (1,3)
prove that the vectors (1,0,0) ( 0,1,0) (1,1,0) is linearly dependent .
Find the inverse of A = ( 1,3,0) ( 0,4,-6) ( -1,5,7) .
Find the basis and dimension of the vectors are (1,-3,1) ( 2,-6,2) and (3,-9,3) .
a) Is the set of vectors {(1,2,3), (3,4,1),(2,3,2)} linearly independent? Give reasons for the answer.
6 4 1 5 14
8 9 2 7 16
4 3 6 2 5
6 10 15 4
b) Find an initial basic feasible solution to the following transportation problem by the North-West corner method. Verify whether your solution is optimal.
Suppose T 2 L(R2) is deÖned by T(x; y) = ((3y; x). Find the eigenvalues of T
Prove that there does not exist a linear map T : R5 ! R5
such that range T = null T.
Suppose S; T 2 L(V ) are such that ST = T S. Prove that null S is invariant under T.
