1. ENGG2013 Unit 6 Matrix in action Jan, 2011.

2. Linear transformation A.k.a. Linear mapping, linear function. A way to map an m-dimensional object to an n-dimensional object. kshumENGG20132 3-D to 2-D transformation 2-D to 3-D transformation

3. Historical note Matrix algebra was developed by Arthur Cayley (1821~1895) – Memoir on the theory of matrices (1858) The term “matrix” was coined by James Joseph Sylvester (1814~1897) in 1850. kshumENGG20133 http://en.wikipedia.org/wiki/James_Joseph_Sylvester http://en.wikipedia.org/wiki/Arthur_Cayley

4. Today’s objective kshumENGG20134 Why do we define matrix multiplication in such a strange way?

5. Matrix as action Matrix-vector product is a function from a vector space to another vector space. kshumENGG20135 Multiply by M v M v

6. Review of function in mathematics A function consists of – Domain: a set – Range: another set – An association between the elements. kshumENGG20136 Domain Range x f(x)

7. Example kshumENGG20137 Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E The function L L(Boy 1) = Girl A L(Boy 2) = Girl C, Etc. “L” stands for “love” Domain Range

8. An ideal case kshumENGG20138 Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E One-to-one function a.k.a. injective function Domain Range

9. Question kshumENGG20139 Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E DomainRange How many possible functions can we make? How many of them are one-to-one?

10. Example 1 Reflection Domain: Range: Define kshumENGG201310

11. Example 2 Rotation by 90 degrees Domain: Range: Define kshumENGG201311

12. Example 3 Projection Domain: Range: Define kshumENGG201312 No. of input varaibles No. of output variables

13. Example 4 Domain: Range: Define a function kshumENGG201313

14. Cascading two functions kshumENGG201314 multiply by  3 Rotate 90 degrees and scale up by a factor of 3. Example:

15. Function composition Can we compose the functions in example 3 and example 4 and do the computation in one step? kshumENGG201315 multiply by ?

16. More generally… Can you repeat the same thing for any two matrices and ? kshumENGG201316 multiply by ?

17. Even more generally kshumENGG201317 multiply by A multiply by B multiply by ? u v w uw A is m x n, B is n x p What goes in here is the matrix product A B You can find the definition of two matrices in any textbook on linear algebra, or from the web.

18. Main points Matrix-vector multiplication is an action. – It is useful in computer graphics and geometry. “Matrix time matrix” is the same as function composition. The definition of the product of two matrices follows naturally from this viewpoint. kshumENGG201318