Chapter 2: Matrix Algebra

Class Notes

• Lay, Section 2.1, Matrix Operations (pdf)
• Lay, Section 2.2, Matrix Inverse (pdf)
• Lay, Section 2.3, Characterizations of Invertible Matrices (pdf)
  
  
  • Wikipedia entry for "condition number": Given the equation Ax=b, how much change in x will result from a small change in b?
  
  
• Lay, Section 2.4, Partitoned Matrices (pdf)
• Lay, Section 2.5, Matrix Factorization (pdf)
• Lay, Section 2.6, Leontief Input-Output Model (pdf)
  
  
  • Autobiography of Wassily Leontief (1906-1999)
  • Nobel Prize lecture, 1973
  • Short biography and major works
  • Pronounciation of "Leontief" (click on the megaphone icon)
  
  
• Lay, Section 2.7, Applications to Computer Graphics (pdf)
• Lay, Section 2.8, Subspaces of Rⁿ (pdf)
• Lay, Section 2.9, Dimension and Rank (pdf)

Homework Exercises

• Lay, Section 2.1, pp 116--118 -- 3, 5, 7, 10, 14 (cost matrix), 35 (special matrices), 36 (random matrices), 39 (nilpotent matrix), 40 (powers of a matrix)
• Lay, Section 2.2, pp 126--128 -- 3, 6, 15, 17, 21, 29, 31, 35, 39, 40 (stiffness matrix), 41 (elastic beam), 42 (deflection)
• Lay, Section 2.3, pp 132--134 -- 1, 3, 5, 11, 17, 19, 23, 33, 41 (condition number), 44--45 (Hilbert matrices, condition numbers, and the accuracy of solutions)
• Lay, Section 2.4, pp 139--141 -- 5, 10, 15 (Schur complement), 17 (deep space probe), 23--24 (induction arguments), 25 (inverse of a block diagonal matrix)
• Lay, Section 2.5, pp 149--151 -- 2, 7, 15, 17, 24 (QR factorization), 25 (singular value decomposition), 29 (ladder network), 31 (band matrices), 32 (tridiagonal matrices and heat conduction in a rod)
• Lay, Section 2.6, pp 156--157 -- 1--4 (Leontief Model for a three sector economy), 5, 7, 11 (two expressions for the gross domestic product), 13 (US economy in 1958), 15 (iterative solutions)
• Lay, Section 2.7, pp 165--167 -- 1, 2, 3, 7, 14, 15, 17, 19 (perspective projection)
• Lay, Section 2.8, pp 173--175 -- 1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 22, 23, 31--36, 37 (bases for column space and null space of A)
• Lay, Section 2.9, pp 180--182 -- 1, 3, 7, 8, 9, 15, 17--26, 29, 30 (coordinate vectors)

Review

Use the following exercises to review and test your understanding of the material in Chapter 2. This is not homework, and there is nothing to hand in from this activity.

• Lay, Supplementary Exercises, Chapter 2, pp. 183--184 -- 2, 3, 4, 6, 8, 9, 10, 11 (Vandermonde matrices and interpolating polynomials), 12, 13, 14 (Householder reflection), 19, 20

These additional review exercises from Lay Chapter 2 focus on more theoretical aspects of this material. They generally require proofs or a more sophisticated understanding of the theory of linear algebra.