Download ENGG2013 Lecture 6 - Chinese University of Hong Kong

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
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• The term “matrix” was coined
by James Joseph Sylvester
(1814~1897) in 1850.
http://en.wikipedia.org/wiki/Arthur_Cayley
http://en.wikipedia.org/wiki/James_Joseph_Sylvester
Historical note
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• Matrix algebra was developed
by Arthur Cayley (1821~1895)
– Memoir on the theory of matrices (1858)
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Today’s objective
Why do we define
matrix multiplication
in such a strange way?
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Matrix as action
• Matrix-vector product is a function from a
vector space to another vector space.
v
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Multiply by M
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Mv
5
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
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Example
Domain
The function L
L(Boy 1) = Girl A
L(Boy 2) = Girl C,
Etc.
Range
Boy 1
Girl A
Boy 2
Girl B
Boy 3
Girl C
Boy 4
Girl D
Boy 5
Girl E
“L” stands for “love”
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An ideal case
Domain
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One-to-one function
a.k.a. injective function
Range
Boy 1
Girl A
Boy 2
Girl B
Boy 3
Girl C
Boy 4
Girl D
Boy 5
Girl E
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Question
Domain
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How many possible functions
can we make?
How many of them are
one-to-one?
Range
Boy 1
Girl A
Boy 2
Girl B
Boy 3
Girl C
Boy 4
Girl D
Boy 5
Girl E
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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
No. of
input varaibles
No. of output
variables
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Example 4
• Domain:
• Range:
• Define a function
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Cascading two functions
Example:
multiply by

3
Rotate 90 degrees and scale up by a factor of 3.
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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
?
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More generally…
• Can you repeat the same thing for any two
matrices
and
?
multiply by
multiply by
multiply by
?
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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
?
What goes in here
is the matrix product A B
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You can find
the definition
of two matrices
in any textbook
on linear algebra,
or from the web.
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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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