Vector Spaces
(ppt):
|
43
| - Spanning the Space
| ∀V,
∀⟨w1...wd⟩,
∀v∈V,
∃⟨c1...cd⟩,
v=c1w1+...+cdwd
|
| - Linear Transformations
| A matrix times the vector
⟨c1...cd⟩
for one v gives the vector
⟨c'1...c'd⟩
for another v'
|
| - Change of Basis
| A matrix times the vector
⟨c1...cd⟩
given one basis
⟨w1...wd⟩
gives
|
| |
the
⟨c'1...c'd⟩
for another
⟨w'1...w'd⟩
|
| One proof & many applications:
|
| - Colour
| Red/Blue/Green forms a basis for humans, but in science light is
infinitely dimensional
|
| - Error Correcting Codes
| Encoding is a linear transformation from
the message to the code vectors
|
| - Fourier Transformation
| A change from the sine basis to the time basis
|
| - Integer Multiplication
| A change from the polynomial basis to the evaluation basis
|
| - JPEG Image Compression
| A change from the pixel basis to the sine basis
|
| - Dim Reduce & Face Recog
| A change from xy basis to rotated basis
|
| - Calculus
| Integrating is the inverse linear transformation of differentiating
|
| - Probabilistic Markov Process
| Each time step is a linear transformation of the probability vector
|
| - Quantum Probability
| Each time step is a different linear transformation of the
probability vector
|
Information Theory Example
(ppt):
|
| Knowing Mud Problem
| I know that my mom knows that my dad know that I am good.
|
| Common Knowledge
| Everybody knows that everybody knows that everybody knows
|
| Defn of "Knowing"
| It is true in every universe that I consider possible
|
| |