Review_VectorsMatrices.ppt - Linear Algebra A gentle introduction Linear Algebra has become as basic and as applicable as calculus and fortunately it is

Review_VectorsMatrices.ppt - Linear Algebra A gentle...

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Yulia Peet Arizona State University 1 : “ shiv rpi Linear Algebra A gentle introduction Linear Algebra has become as basic and as applicable as calculus, and fortunately it is easier. --Gilbert Strang, MIT
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Yulia Peet Arizona State University 2 : “ shiv rpi What is a Vector ? Think of a vector as a directed line segment in N-dimensions ! (has “length” and “direction”) Basic idea: convert geometry in higher dimensions into algebra! Once you define a “nice” basis along each dimension: x-, y-, z-axis … Vector becomes a 1 x N matrix! v = [a b c] T Geometry starts to become linear algebra on vectors like v ! c b a v x y v
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Yulia Peet Arizona State University 3 : “ shiv rpi Vector Addition: A+B A B A B C A+B = C (use the head-to-tail method to combine vectors) A+B
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Yulia Peet Arizona State University 4 : “ shiv rpi Scalar Product: a v ) , ( ) , ( 2 1 2 1 ax ax x x a a v v v av av Change only the length (“scaling”), but keep direction fixed . Sneak peek: matrix operation ( Av ) can change length, direction and also dimensionality !
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Yulia Peet Arizona State University 5 : “ shiv rpi Inner (dot) Product: v.w or w T v v v w w 2 2 1 1 2 1 2 1 . ) , ).( , ( . y x y x y y x x w v The inner product is a The inner product is a SCALAR! SCALAR! cos || || || || ) , ).( , ( . 2 1 2 1 w v y y x x w v w v w v 0 .
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  • Fall '19
  • Yulia Peet

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