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Understanding Linear Transformations and Matrix Multiplication in Geometry

This resource explores the principles of linear transformations, also known as linear mappings or functions, which are used to convert m-dimensional objects into n-dimensional objects, such as 2D to 3D and vice versa. We will examine the mathematical foundations laid by Arthur Cayley and James Joseph Sylvester, highlighting how matrix algebra and multiplication function in a geometric context. The document also includes practical examples and questions to deepen understanding of matrix-vector products and function compositions, as applied in fields like computer graphics.

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Understanding Linear Transformations and Matrix Multiplication in Geometry

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  1. ENGG2013Unit 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. 2-D to 3-D transformation 3-D to 2-D transformation ENGG2013

  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. http://en.wikipedia.org/wiki/James_Joseph_Sylvester http://en.wikipedia.org/wiki/Arthur_Cayley ENGG2013

  4. Today’s objective Why do we definematrix multiplicationin such a strange way? ENGG2013

  5. Matrix as action • Matrix-vector product is a function from a vector space to another vector space. Multiply by M v M v ENGG2013

  6. Review of function in mathematics • A function consists of • Domain: a set • Range: another set • An association between the elements. Range Domain f(x) x ENGG2013

  7. Example The function LL(Boy 1) = Girl A L(Boy 2) = Girl C, Etc. Domain Range Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E “L” stands for “love” ENGG2013

  8. An ideal case One-to-one functiona.k.a. injective function Domain Range Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E ENGG2013

  9. Question How many possible functionscan we make?How many of them are one-to-one? Domain Range Boy 1 Boy 2 Boy 3 Boy 4 Boy 5 Girl A Girl B Girl C Girl D Girl E ENGG2013

  10. Example 1 Reflection • Domain: • Range: • Define ENGG2013

  11. Example 2 Rotation by 90 degrees • Domain: • Range: • Define ENGG2013

  12. Example 3 Projection • Domain: • Range: • Define No. ofinput varaibles No. of outputvariables ENGG2013

  13. Example 4 • Domain: • Range: • Define a function ENGG2013

  14. Cascading two functions Example: multiply by  3 Rotate 90 degrees and scale up by a factor of 3. ENGG2013

  15. Function composition • Can we compose the functions in example 3 and example 4 and do the computation in one step? multiply by multiply by multiply by ? ENGG2013

  16. More generally… • Can you repeat the same thing for any two matrices and ? multiply by multiply by multiply by ? ENGG2013

  17. Even more generally multiply by B u multiply by A w v A is m x n, B is n x p multiply by u w You can findthe definitionof two matricesin any textbookon linear algebra,or from the web. ? What goes in hereis the matrix product A B ENGG2013

  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. ENGG2013

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