(1) Find the rank of A.
(2) Find the value of the determinant A.
(1) Find the inverse of A.
(2) Solve the equation AX = B .
Let the subspace V be generated by
v1 = (1,1,0,1) , v2 = (2,0,0,1) , v3 = (1,3,0,2) ,
and the subspace W be generated by
w1 = (1,2,3,0) , w2 = (0,1, 2,3) , w3 = (0,0,1,2) .
(1) Find the dimension of V ∩W .
(2) Find the dimension of V +W .
T : R2 → R3 is a linear mapping defined by
T(x1, x2 ) = (x1 + 2x2 , 7x1 + 3x2 , 2x1 + x2 ) .
(1) Show that T is one to one.
(2) Show that T maps R2 onto R3.
(1) Find the rank of T.
(2) Find a basis for the range of T.
(3) Find a basis for the kernel of T.
(1) Find the eigenvalues of A.
(2) For each eigenvalue of A , find a eigenvector of A corresponding to this
eigenvalue.
respect to the standard basis of R3 .
(1) Find the characteristic values (eigenvalues) of T.
(2) Show that T is diagonalizable or not.
diagonal.
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