Linear Algebra Answers

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

Let T 2 LLR3
such that 4, 5 and p7 are its eigenvalues.

Show that T(x)9x = ((4;5; p7).

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.

Suppose V is Önite-dimensional with dim V  2. Prove that there exist S; T 2 L(V; V ) such that ST 6= T S.

Suppose b,c 2 R, and T: R3 ! R2 deÖned as

T (x; y; z) = (2x x 4y + 3z + b; 6x + cxy):

Show that T is linear if and only if b = c = 0

Suppose V and W are Önite-dimensional and T 2 L(V; W). Show that with respect to each choice of bases of V and W, the matrix of T has atl east dim range T nonzero entries.