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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