Jan19-12

Jan19-12 - Math 260, Spring 2012 Jerry L. Kazdan Class...

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Math 260, Spring 2012 Jerry L. Kazdan Class Outline: Jan. 19, 2012 1. Definitions: Homogeneous equation, inhomogeneous equation. Basic Lemma: If you have n linear algebraic equations in k unknowns, and if n > k (so more equations than unknowns), then the homogeneous equations always have at least one non-trivial solution ( nontrivial means a solution other than 0). In other words, the dimension of the nullspace of the corresponding map is greater than 0. Definitions: For a linear map L : V W (here V and W are linear spaces): The image of L is the set of all y W for which Lx = y has a solution. The nullspace or kernel of L is the set of all solutions of the homogeneous equation Lx = 0. Notation: I ( L ), N ( L ), ker(L). 2. Maps: f : S → T . Definitions: one-to-one (injective), onto (surjective), invertible (bijective) 3. A : R 2 R 2 If A is linear, it maps straight lines to straight lines, and preserves parallelism Step 0: What is a straight line? Step 1: What about degeneracies like
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Jan19-12 - Math 260, Spring 2012 Jerry L. Kazdan Class...

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