1. Introductory Linear Algebra Erasmus Mobility Program (24Apr2012) Pollack Mihály Engineering Faculty (PMMK) University of Pécs João Miranda jlmiranda@estgp.pt http://tiny.cc/jlm_estg ; http://tiny.cc/jlm_ist

2. 24-Apr-2012Introductory LA2 Vectors e vector calculus; Matrix and matrix calculus; Determinants; Linear Systems of Equations.

3. 24-Apr-2012Introductory LA3 A furniture factory builds Tables ( t ) at a profit of 4 Euros per Table, and Chairs ( c ) at a profit of 3 Euros per Chair). Suppose that only 8 short ( s ) pieces and 6 large ( l ) pieces are available for building purposes, what combination of Tables and Chairs do you need to build to make the most profit?. If the availability of the short pieces is 8008 and the availability of the large pieces is 6007, how many Tables and Chairs do you need to build to make the most profit? Tables & Chairs

4. 24-Apr-2012Introductory LA4 Note: modeling is based in the proportionality, aditivity and divisibility between the produced quantities of t and c and profit and utilization of l and s components.. Tables & Chairs

5. 24-Apr-2012Introductory LA5 Note: modeling is based in the proportionality, aditivity and divisibility between the produced quantities of t and c and profit and utilization of l and s components. Tables & Chairs

6. 24-Apr-2012Introductory LA6 Tables & Chairs To maximize the profit function, Prft(t,c), but do not overpass the availability of components l and s.. Problem: –The maximum of profit is targeted, –The decision variables t and c are constrained by the availabilities of l and s:

7. 24-Apr-2012Introductory LA7 By Gauss Elimination, below main diagonal: By ascent substitution : Gauss Elimination

8. 24-Apr-2012Introductory LA8 By Gauss Elimination, the correct procedure is : a) To multiply line 1 by (3/2) and subtract it to line 2; b) To divide line 1 by 2 and multiply it by (-3); then, add it to line 2; c) Divide line 1 by 2 and multiply it by (-3); then, subtract it to line 2; d) Both alternatives, a and b. Q) Gauss Elimination

9. 24-Apr-2012Introductory LA9 Applying Gauss Elimination (to the augmented matrix): By ascent substitution : Gauss Elimination

10. 24-Apr-2012Introductory LA10 Elimination by Gauss-Jordan drives the inverse matrix : Gauss-Jordan Inverse Matrix

11. 24-Apr-2012Introductory LA11 For the presented system, classify the following statements (T/F): 1) The system has the trivial solution (0,0). 2) The system has one and only one solution. 3) The system has infinitely many solutions. Q) Gauss Elimination

12. 24-Apr-2012Introductory LA12 Using the inverse matrix, Inverse Matrix

13. 24-Apr-2012Introductory LA13 From the inverse matrix, if … 1) The first RHS parameter grows 1, the first variable grows… ? 2) The first RHS parameter grows 1, the second variable grows… ? Q) Inverse Matrix

14. 24-Apr-2012Introductory LA14 Applying the Cramer’s Rule to the linear system of equations, Cramer’s Rule

15. 24-Apr-2012Introductory LA15 Applying the Cramer’s Rule to the presented system of equations, it is true the alternative in : a) t = 2003; c = 2002; b) t = 2004; c = 2001; c) The system has no solution; d) None of the above. Q) Cramer’s Rule

16. 24-Apr-2012Introductory LA16 Introductory LA (synthesis) Vectors e vector calculus; Matrix and matrix calculus; Determinants; Linear Systems of Equations.