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Review_I - Lecture Note Review I Oct 10 Dr Jeff Chak-Fu...

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Lecture Note: Review I: Oct 10 - Oct 13, 2006 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2006 Produced by Jeff Chak-Fu WONG 1
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L INEAR E QUATIONS AND M ATRICES L INEAR E QUATIONS AND M ATRICES 2
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1. Method of elimination. To solve a linear system, repeatedly perform the following operations: (a) Interchange two equations . (b) Multiply an equation by a non-zero constant . (c) Add a multiple of one equation to another equation . 2. Matrix Operations. (a) Addition (b) Scalar Multiplication (c) Transpose (d) Multiplication 3. Theorem 0.1 : Properties of matrix addition. 4. Theorem 0.2 : Properties of matrix multiplication. 5. Theorem 0.3 : Properties of scalar multiplication. 6. Theorem 0.4 : Properties of transpose. 7. Reduced row echelon form. 8. Procedure for transforming a matrix to reduced row echelon form. L INEAR E QUATIONS AND M ATRICES 3
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9. Gauss-Jordan reduction procedure (for solving the linear system A x = b ). 10. Gauss elimination procedure (for solving the linear system A x = b ). 11. Theorem 0.8 : A homogeneous system of m equations in n unknowns always has a nontrivial solution if m < n . 12. Theorem 0.10 : Properties of the inverse. 13. Practical method for finding A - 1 14. Theorem 0.12 : An n × n matrix is non-singular if and only if it is row equivalent to I n . 15. Theorem 0.13 : If an n × n matrix, the homogeneous system A x = 0 has a nontrivial solution if and only if A is singular. L INEAR E QUATIONS AND M ATRICES 4
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16. List of Nonsingular Equivalences : The following statements are equivalent. (a) A is nonsingular. (b) x = 0 is the only solution to A x = 0 . (c) A is row equivalent to I n . (d) The linear system A x = b has a unique solution for every n × 1 matrix b .
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