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Unformatted text preview: MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1. Review: Linear Algebra I Linear systems of equations 1 1.1. Definition 1 1.2. Solving linear systems of equations 1 2. Linear Algebra II Vectors and matrices 1 2.1. Definition 1 2.2. Operations 2 2.2.1. Basic Matrix Operations 2 2.2.2. Matrix-Vector multiplication 4 2.2.3. Matrix-matrix multiplication 5 2.3. Summary of Matrix Operation Laws: [reading material] 8 1. Review: Linear Algebra I Linear systems of equations 1.1. Definition. 1.2. Solving linear systems of equations. The Gaussian Elimination Algorithm Reduced row-echelon form 2. Linear Algebra II Vectors and matrices 2.1. Definition. Definition. A matrix is a rectangular array of numbers A = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a m 1 a m 2 a mn The elements a ij of the matrix A are called entries . If the matrix has m rows and n columns, it is called an m n matrix. Note: If the size of the matrix A is clear, we can use the shorthand notation A = [ a ij ]. If m = n , then A is a square matrix . eg. a 3 3 square matrix: - 1 3 1- 1 5 4 3 If A is a square matrix, then the elements a ii are called the diagonal elements and their sum are called the trace of A . Date : 2010-03-08. 1 2 JING LI For the above example, the diagonal elements are- 1 , 1 , 3, and the trace is tr( A ) =- 1+1+ 3 = 3. A 1 n matrix is called a row vector : [ c 1 ,c 2 , ,c n ]. eg. a 1 4 row vector: [1 3 0 5]. An m 1 matrix is called a column vector : b 1 b 2 . . . b m . eg. a 3 1 column vector: 2 7 4 . Two matrices A = [ a ij ] ,B = [ b ij ] are said to be equal if they have the same dimension and if for all i,j , we have a ij = b ij . the zero matrix : . . . . . . . . . , the size of this matrix is usually clear from the context. the identity matrix : 1 0 0 1 . . . . . . . . . . . . 0 0 1 , the size of this matrix is usually clear from the context....
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