## Mathematics 210## Linear Algebra |

* Chapter 4: Vector Spaces *

- Lay, Section 4.1, Vector Spaces and Subspaces (pdf)
- Lay, Section 4.2, Null Spaces, Column Spaces, and Linear Transformations (pdf)
- Lay, Section 4.3, Linearly Independent Sets; Bases (pdf)
- Lay, Section 4.4, Coordinate Systems (pdf)
- Lay, Section 4.5, The Dimension of a Vector Space (pdf)
- Lay, Section 4.6, Rank (pdf)
- Lay, Section 4.7, Change of Basis (pdf)
- Lay, Section 4.8, Applications to Difference Equations (pdf)
- Biography and photos of Felice Casorati (1835-1890)

- Lay, Section 4.9, Applications to Markov Chains (pdf)
- Biography and photos of Andrei Andreyevich Markov (1856-1922)

*Lay, Section 4.1, pp 223--225 --*5, 7, 9, 15, 19, 20, 21, 31, 32 (intersection of subspaces), 33, 34 (H+K)*Lay, Section 4.2, pp 234--236 --*1, 3, 9, 15, 19, 23, 33, 34, 39, 40 (intersection of two subspaces)*Lay, Section 4.3, pp 243--245 --*3, 9, 12, 13, 19, 37, 38 (sines and cosines)*Lay, Section 4.4, pp 253--255 --*1, 5, 9, 11, 13, 27, 37, 38 (crystal lattice for titanium)*Lay, Section 4.5, pp 260--262 --*3, 7, 9, 11, 15, 21, 23 (Hermite polynomials), 27 (polynomials), 33 (expanding a linearly independent set to a basis), 34 (cosines)*Lay, Section 4.6, pp 269--271 --*1, 3, 5, 7, 11, 15, 27, 28, 29, 30, 35 (fundamental subspaces)*Lay, Section 4.7, pp 276--277 --*1, 3, 5, 7, 9, 17, 18 (cosines)*Lay, Section 4.8, pp 285--287 --*1, 3, 5, 9, 13, 19, 20 (cantilevered beam), 21 (moving average), 29 (first-order system)*Lay, Section 4.9, pp 296--298 --*1, 4, 11, 14, 15 (California demographics), 16 (Hertz rental cars), 17, 18, 19 (stochastic matrices)

Download the supporting pdf file and Mathematica notebook for the following project from the Lay Linear Algebra web site.

*Space Flight and Control Systems*

The description of the following project, and its supporting Mathematica notebook, are available from David Smith's Connected Curriculum Project at Duke University.

Use the following exercises to review and test your understanding of the material in Chapter 4. This is not homework, and there is nothing to hand in from this activity.

*Lay, Supplementary Exercises, Chapter 4, pp. 298-300 --*12, 13, 14, 15 (rank theorems), 16 (rank factorization), 18, 19, 21 (controllability matrices)

These additional review exercises from Lay Chapter 4 focus on more theoretical aspects of this material. They generally require proofs or a more sophisticated understanding of the theory of linear algebra.