1. ERASMUS+ Zintegrowane nauczanie przedmiotów ścisłych z perspektywą obliczeniową The program was financed by the EU - ESF

2. Workshops 2 SAGE

3. Linear Algebra Sage provides standard constructions from linear algebra, e.g., the characteristic polynomial, echelon form, trace, decomposition, matrix operatiing. Creation of matrices and matrix multiplication is easy and natural: Projekt finansowany ze środków EFS

4. Examples sage: A = Matrix([[1,2,3],[3,2,1],[1,1,1]]) sage: w = vector([1,1,-4]) sage: w*A(0, 0, 0) sage: A*w(-9, 1, -2) sage: kernel(A) Free module of degree 3 and rank 1 over Integer RingEchelon basis matrix:[ 1 1 -4] Projekt finansowany ze środków EFS

5. Examples Solving matrix equations is easy, using the method solve_right. Evaluating A.solve_right(Y) returns a matrix (or vector) X so that AX=Y: sage: Y = vector([0, -4, -1]) sage: X = A. solve_right(Y) sage: X(-2, 1, 0) sage: A * X # checking our answer...(0, -4, -1) Projekt finansowany ze środków EFS

6. Linear Algebra A backslash \ can be used in the place of solve_right; use A \ Y instead of A.solve_right(Y). sage: A \ Y(-2, 1, 0) If there is no solution, Sage returns an error: sage: A.solve_right(w) Traceback (most recent call last):...ValueError: matrix equation has no solutions Projekt finansowany ze środków EFS

7. Linear Algebra Similarly, use A.solve_left(Y) to solve for X in XA=Y. Sage can also compute eigenvalues and eigenvectors: sage: A = matrix([[0, 4], [-1, 0]]) sage: A.eigenvalues ()[-2*I, 2*I] sage: B = matrix([[1, 3], [3, 1]]) sage: B.eigenvectors_left() [(4, [(1, 1)], 1), (-2, [(1, -1)], 1)] Projekt finansowany ze środków EFS

8. Exercises for students Please verify that the system is a Cramer. If so, solve it using determinants. Projekt finansowany ze środków EFS