# Lecture 14 - Fundamentals For Systems of ODEs Eigenvalues...

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Fundamentals For Systems of ODEs Eigenvalues, Eigenvectors, and Linear Independence

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Last Time On Wednesday, we established the rules of matrix algebra. a 11 a 21 a 12 a 22 x 1 x 2 = a 11 x 1 a 12 x 2 a 21 x 1 a 22 x 2 + + This definition is the foundation for everything we did.
Can View Matrices as Mapping Vectors to Vectors a 11 a 21 a 12 a 22 x 1 x 2 = y 1 y 2 2 4 - 1 1 2 = 0 - 3 - 2

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Can View Matrices as Mapping Vectors to Vectors 2 4 - 1 1 2 = 0 - 3 - 2 Remember, we can represent vectors as arrows on a plane ( 1 , 2 )
Can View Matrices as Mapping Vectors to Vectors 2 4 - 1 1 2 = 0 - 3 - 2 ( 0 , - 2 )

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A x = λ x We Are Interested In A Special Case
A x = λ x We Are Interested In A Special Case

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A x = λ x We Are Interested In A Special Case
A x = λ x We Are Interested In A Special Case

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A x = λ x 2 4 - 1 = 1 4 - 3 - 2 - 8 We Are Interested In A Special Case
A x = λ x 2 4 - 1 = 1 4 - 3 - 2 - 8 We Are Interested In A Special Case

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A x = λ x 2 4 - 1 = 1 4 - 3 - 2 - 8 = - 2 1 4 These are the same We Are Interested In A Special Case
A x = λ x Eigenvector Eigenvalue We Are Interested In A Special Case

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But How Do We Find Eigenvalues and Eigenvectors?
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