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Unformatted text preview: BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Linear Algebra BME 5020 Computer and Mathematical Application in Bioengineering © 2006 Jingwen Hu September 28, 2006 • Gauss Elimination • LU Decomposition • Matrix Inverse Contents BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Introduction BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University • An equation of the form ax+by+c=0 or equivalently ax+by=c is called a linear equation in x and y variables. • ax+by+cz=d is a linear equation in three variables, x, y , and z . • Thus, a linear equation in n variables is a 1 x 1 +a 2 x 2 + … +a n x n = b • A solution of such an equation consists of real numbers c 1 , c 2 , c 3 , … , c n . If you need to work more than one linear equations, a system of linear equations must be solved simultaneously. BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Introduction BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University • A system of equations with the form: n n nn n n n n n n b x a x a x a b x a x a x a b x a x a x a = + + + = + + + = + + + ... ... ... ... ... 2 2 1 1 2 2 2 22 1 21 1 1 2 12 1 11 is typically referred to as linear algebraic equations. Frequently, these equations are expressed as [A] {x} = {b} BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix • Symmetric matrix: a ij = a ji BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 8 7 2 7 3 1 2 1 5 ] [ A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix • Diagonal matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 44 33 22 11 ] [ a a a a A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix • Identity matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 1 1 1 1 ] [ I BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix • Upper triangular matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 44 34 33 24 23 22 14 13 12 11 ] [ a a a a a a a a a a A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix • Lower triangular matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 44 43 42 41 33 32 31 22 21 11 ] [ a a a a a a a a a a A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix • Banded matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 44 43 34 33 32 23 22 21 12 11 ] [ a a a a a a a a a a A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University...
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 Fall '06
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 Biomedical Engineering, Determinant, Triangular matrix, Howard Staunton, Wayne State University

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