ENGG2013 Unit 6 Matrix in action
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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.
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3-D to 2-D transformation2-D to 3-D transformation
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.
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Today’s objective
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Why do we definematrix multiplication
in such a strange way?
Matrix as action
• Matrix-vector product is a function from a vector space to another vector space.
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Multiply by Mv M v
Review of function in mathematics
• A function consists of – Domain: a set– Range: another set– An association between the elements.
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DomainRange
x f(x)
Example
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Boy 1
Boy 2
Boy 3
Boy 4
Boy 5
Girl A
Girl B
Girl C
Girl D
Girl E
The function LL(Boy 1) = Girl AL(Boy 2) = Girl C,Etc.
“L” stands for “love”
Domain Range
An ideal case
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Boy 1
Boy 2
Boy 3
Boy 4
Boy 5
Girl A
Girl B
Girl C
Girl D
Girl E
One-to-one functiona.k.a. injective functionDomain Range
Question
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Boy 1
Boy 2
Boy 3
Boy 4
Boy 5
Girl A
Girl B
Girl C
Girl D
Girl E
Domain Range
How many possible functionscan we make?How many of them are one-to-one?
Example 1 Reflection
• Domain:• Range:• Define
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Example 2 Rotation by 90 degrees
• Domain:• Range:• Define
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Example 3 Projection
• Domain: • Range:• Define
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No. ofinput varaibles
No. of outputvariables
Example 4
• Domain:• Range:• Define a function
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Cascading two functions
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multiply by
3
Rotate 90 degrees and scale up by a factor of 3.
Example:
Function composition
• Can we compose the functions in example 3 and example 4 and do the computation in one step?
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multiply by
multiply by
multiply by
?
More generally…
• Can you repeat the same thing for any two matrices and ?
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multiply by
multiply by
multiply by
?
Even more generally
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multiply by Amultiply by B
multiply by
?
u
v
w
u w
A is m x n,B is n x p
What goes in hereis the matrix product A B
You can findthe definitionof two matricesin any textbookon linear algebra,or from the web.
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.
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