Ch7-Linear-Algebra copy

# Ch7-Linear-Algebra copy - Review of Linear Algebra Chapter...

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Unformatted text preview: 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 z scalar – lowercase Greek ( V¡D¡E ) z vector – lowercase roman (u,v,x,y,b) z matrix – uppercase roman (A,B,C) » » » » ¼ º « « « « ¬ ª m x x x x ¡ 2 1 > @ n y y y y ¢ 2 1 2 Vector Operations z Addition and Subtraction involve corresponding elements of a different vector with the same number of elements z Multiplication by a Scalar involves multiplication of every element by the scalar z Vector Transpose converts row vector to a column vector or vice versa 3 Linear Combination involves scalar multiplication and vector addition w v u ¡ E D 1 1 1 1 1 2 2 2 2 2 m m m m m u v u b v w u v u b v w u v u b v w D D D E D ¡ ª º ª º ª º ª º « » « » « » « » ¡ « » « » « » « » ¡ « » « » « » « » « » « » « » « » ¡ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¡ ¡ ¡ ¡ 4 Vector Inner Product is an operation between two vectors that have the same number of elements z ALWAYS results in a scalar z MUST be a row vector times a column vector z Can use transpose for two column vectors ¦ ¡ ¢ n i i i y x y x 1 V V > @ 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 £ £ £ » » » » ¼ º « « « « ¬ ª 5 Vector Norms compare the size (magnitude) of a vector z For example, it makes sense that z “Unit” vector has a magnitude of 1 z “Zero” vector has a magnitude of 0 z Absolute Value is a measure of magnitude for scalars z Norm is a measure of magnitude for vectors E D !...
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Ch7-Linear-Algebra copy - Review of Linear Algebra Chapter...

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