Lecture 13 - Review of Matrices Or A Fast Introduction Why...

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Review of Matrices Or A Fast Introduction
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Why Study Matrices? Fundamental Tools For Linear Systems y 1 0 = a 11 ( t ) y 1 + a 12 ( t ) y 2 + ... + a 1 n ( t ) y n y 2 0 = a 21 ( t ) y 1 + a 22 ( t ) y 2 + ... + a 2 n ( t ) y n y n 0 = a n 1 ( t ) y 1 + a n 2 ( t ) y 2 + ... + a nn ( t ) y n . . . + g 1 ( t ) + g 2 ( t ) + g n ( t )
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Matrices Come From Systems of Equations 3 7 2 4 x x z z + + = = We’ve known how to find x z For different right hand sides - 3 - 1
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Matrices Come From Systems of Equations 3 7 2 4 x x z z + + = 3 7 2 4 x z Matrices are all about what happens with different x z Matrix Vector
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Matrices Come From Systems of Equations 3 7 2 4 + + = Matrices are all about what happens with different 3 7 2 4 = = x z = = 1 - 1 ( - 1) ( - 1) (1) (1) 1 3 1 - 1 1 3
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Matrices Come From Systems of Equations Fundamental Definition: Matrix Vector Multiplication 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 + + (Think of it as a system of equations!) 2 4 - 1 2 6 8 Try it!
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Matrices Come From Systems of Equations Fundamental Definition: Matrix Vector Multiplication 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 + + (Think of it as a system of equations!) 2 4 - 1 2 6 8 = Try it! 2 (6) + 4 (8) - 1 (6) + 2 (8) = 44 10
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Matrices Come From Systems of Equations Fundamental Definition: Matrix Vector Multiplication 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 + + Notation a nm n First Index Is Row m Second Index is Column Rows Columns
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Matrices Come From Systems of Equations Fundamental Definition: Matrix Vector Multiplication 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 + + Notation Second Column a 12 First Row
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Matrices Come From Systems of Equations Other Important Definitions Addition x 1 x 2 = 3 7 - 2 4 1 5 6 8 + x 1 x 2 3 x 1 + 1 x 2 7 x 1 + 5 x 2
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Matrices Come From Systems of Equations Other Important Definitions Addition x 1 x 2 = 3 7 - 2 4 1 5 6 8 + x 1 x 2 3 x 1 + 1 x 2 7 x 1 + 5 x 2 x 1 + x 2 x 1 + x 2 - 2 4 6 8 = =
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Matrices Come From Systems of Equations Other Important Definitions Addition x 1 x 2 = 3 7 - 2 4 1 5 6 8 + x 1 x 2 3 x 1 + 1 x 2 7 x 1 + 5 x 2 + + x 1 + x 2 x 1 + x 2 - 2 4 6 8 = =
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Matrices Come From Systems of Equations
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