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ENGG2013 Unit 6 Matrix in action. Jan, 2011. 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. Historical note.
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ENGG2013Unit 6 Matrix in action Jan, 2011.
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
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
Today’s objective Why do we definematrix multiplicationin such a strange way? ENGG2013
Matrix as action • Matrix-vector product is a function from a vector space to another vector space. Multiply by M v M v ENGG2013
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
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
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
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
Example 1 Reflection • Domain: • Range: • Define ENGG2013
Example 2 Rotation by 90 degrees • Domain: • Range: • Define ENGG2013
Example 3 Projection • Domain: • Range: • Define No. ofinput varaibles No. of outputvariables ENGG2013
Example 4 • Domain: • Range: • Define a function ENGG2013
Cascading two functions Example: multiply by 3 Rotate 90 degrees and scale up by a factor of 3. ENGG2013
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
More generally… • Can you repeat the same thing for any two matrices and ? multiply by multiply by multiply by ? ENGG2013
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
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