The Basis

* Coordinate systems

** Coordinate representation

** Linear Combinations: Lossy compression

* Greedy algorithms for finding a set of generators

** Grow algorithm

** Shrink Algorithm

** When greed fails

*** When greed fails: dominating set

** Minimum spanning forest

*** Minimum spanning forest: spanning

*** Minimum spanning forest: forest

*** Minimum spanning forest: Grow algorithm

*** Minimum spanning forest: Shrink algorithm

** Formulating Minimum Spanning Forest in linear algebra

** Grow algorithms

* Linear Dependence

** Linear Dependence: The Superfluous-Vector Lemma

** Defining linear dependence

** Linear dependence in relation to other questions

*** Linear dependence and null space

*** Linear dependence in Minimum Spanning Forest

** Properties of linear independence: hereditary

** Properties of linear (in)dependence

** Analyzing the Grow algorithm

** Analyzing the Shrink algorithm

* Basis

** Basis: Examples

** Basis: Examples in graphs

** Towards showing that every vector space has a basis

** Computational problems involving finding a basis

* Unique representation

** Uniqueness of representation in terms of a basis

** Uniqueness of representation in terms of a basis: The case of graphs

* Change of basis

** Change of basis: simple example

** Perspective rendering

*** Simplified camera model

*** Even more simplified camera model

*** Camera coordinate system

*** From world point to camera-plane point

*** Converting to pixel coordinates

*** From world coordinates to camera coordinates to pixel coordinates

*** Wiimote whiteboard

*** Removing perspective

*** Camera coordinate system

*** Converting from one basis to another

*** Converting between different basis representations

*** Converting...

*** From pixel coordinates to whiteboard coordinates

*** How to almost

*** Almost computing H

*** w1 (hy3 ,x1 x1 + hy3 ,x2 x2 + hy3 ,x3 ) = hy1 ,x1 x1 + hy1 ,x2 x2 + hy1 ,x3

*** One more equation

*** Once you know H

* Exchange Lemma

** Simplified Exchange Lemma

*** Simplified Exchange Lemma proof

** Exchange Lemma proof

* Proof of correctness of the Grow algorithm for Minimum Spanning Forest

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