Ch7-Linear-Algebra

Ch7-Linear-Algebra - PGE 310 Formulation and Solution in...

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Review of Linear Algebra Chapter 7 of Numerical Methods with MATLAB, Gerald Recktenwald PGE 310: Formulation and Solution in Geosystems Engineering Dr. Balhoff Spring 2011 1 Vector is an ordered set of real (or complex) numbers arranged as a row or column scalar – lowercase Greek (  ) vector – lowercase roman (u,v,x,y,b) matrix – uppercase roman (A,B,C) m x x x x 2 1   n y y y y 2 1 2
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Vector Operations Addition and Subtraction involve corresponding elements of a different vector with the same number of elements Multiplication by a Scalar involves multiplication of every element by the scalar Vector Transpose converts row vector to a column vector or vice versa 3 Linear Combination involves scalar multiplication and vector addition w v u   11 1 1 1 22 2 2 2 mmm m m uvu b v w b v w b v w             4
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Vector Inner Product is an operation between two vectors that have the same number of elements ALWAYS results in a scalar MUST be a row vector times a column vector Can use transpose for two column vectors n i i i y x y x 1  4 4 3 3 2 2 1 1 4 3 2 1 4 3 2 1 y x y x y x y x y y y y x x x x 5 Vector Norms compare the size (magnitude) of a vector For example, it makes sense that “Unit” vector has a magnitude of 1 “Zero” vector has a magnitude of 0 Absolute Value is a measure of magnitude for scalars Norm is a measure of magnitude for vectors  y x 6
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Vector Norms u=3i+4j What is the velocity What is the speed? 22 34 5 u  7 Vector Norms Euclidean Norms in 2D and 3D are geometric lengths 13 3 2 20 2 4 2 2 2 2 b a l l 2 4 20 a l 3 2 13 b l x y 3 4 29 c l x y 29 3 2 4 2 2 2 c l z 2 2D 3D 8
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This note was uploaded on 09/23/2011 for the course PGE 310 taught by Professor Klaus during the Spring '06 term at University of Texas.

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Ch7-Linear-Algebra - PGE 310 Formulation and Solution in...

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