Linear Algebra with Maple  - MTH 362 Fall 2000 > restart;                     # Start with clean slate  > with(linalg):                # this loads the linear algebra package > A:=matrix([[1,2],[3,4]]);    # Define a matrix named "A" > B:=matrix([[2,-3],[1,4]]);   # Define a matrix named "B" > C := A+B;                    # Add the matrices A and B, call it "C" > evalm(C);                    # To see the entries of C type this  > evalm(2*A - 5*B);            # multiplication by a scalar and subtraction > D:=2*A + 7*B;                # The symbol D is reserved in maple. > E:=transpose(C);             # Transpose of a matrix > inverse(C);                  # inverse of a matrix > A&*B;                        # multiplication of matrices in maple                                 #      requires &* instead of * > evalm(%);                    # Here is the answer > v := vector([1 , -3]);       # vectors are column vectors written                                 #      horizontally to save space > A&*v;                        # A matrix times a vector > evalm(%);                    # Recall that the answer is a column vector > evalm(v&*A);                 # Maple considers a vector multiplying on                                 # the left as a row vector > a:=vector([1,0,2]); > b:=vector([-2,1,2]); > innerprod(a,b);              # the inner product of the vectors a and b > evalf(angle(a,b));           # The angle between a and b (in radians) > norm(a,2);                   # The magnitude of the vector a  > U := matrix([[1,1,2],[0,4,3],[-1,-2,-3]]):  > augment(U,a);                # The matrix U augmented with the vector a > det(U);                      # The determinant of U > trace(U);                    # The trace of U > rank(U);                     # The rank of U > inverse(U);                  # The inverse of U  > delrows(U,3..3);             # Delete row 3 from U > rowspace(U);                 # A basis for the row space of U > x:=linsolve(U,a);            # Solve the system U x = a > evalm(U&*x-a);               # Check the answer to previous problem > U1 := augment(U,a);          # Augmented matrix > U2:=gausselim(U1);           # Gaussian elimination                                 #     - reduction to upper triangular form > x1:=backsub(U2);             # Solve upper triangular system                                 #       by back substitution > evalm(A);                    # Recall the matrix A > eigenvals(A);                # eigenvalues of A > eigenvects(A);               # eigenvectors of A > eigenvects([[1,1],[0,1]]);   # An eigenvalue that repeats,                                 #      and only one associated eigenvector > eigenvects([[2,1],[-1,2]]);  # complex eigenvalues and eigenvectors