15. Matrix Calculation

15. Matrix Calculation - ENGR 101, Section 100 Matrix...

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ENGR 101, Section 100 M. Wellman 1 Matrix Calculations ENGR 101, Lecture 15: 8 Nov 10 Background Survey How much do you already know about matrix algebra? A. Good facility with linear algebra B. Some basic matrix operations (e.g., multiplication) C. Have heard of matrices D. Nothing at all
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ENGR 101, Section 100 M. Wellman 2 Announcements Exam 2 Results Median: 68 Project 6 due Wed 10 Nov 9pm Matrices Matrix: a two-dimensional array of values rows and columns of values, of uniform type can be represented in C++ as a vector of vectors Matrices of numbers commonly used in engineering calculations 3 2 5 4 4 1 0 2 –1
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ENGR 101, Section 100 M. Wellman 3 Matrix Elements The element of matrix A in row i , column j is labeled A ij . m × n matrix: m rows, n columns square matrix: same number of rows and columns ( m = n ) A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 A = B 11 B 12 B 13 B 21 B 22 B 23 B 31 B 32 B 33 B 41 B 42 B 43 B = Vectors as Matrices A vector with n elements can be viewed as a 1 × n matrix: A column vector with n elements is an n × 1 matrix: b 1 b 2 b 3 b n – 1 b n b = b 1 b 2 b n – 1 b n b T = Transpose operator ( T ): Converts rows to columns and vice versa
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ENGR 101, Section 100 M. Wellman 4 Multiply Vectors To multiply a row vector by a column vector: (must be equal lengths) multiply them element by element and sum the result result is a scalar a 1 a 2 a n – 1 a n b 1 b 2 b n – 1 b n = a i i = 1 n b i Multiple Choice A. 500 B. 600 C. 120,000,000 D. [ 10 20 30 40 500 ] E. [ 1,000,000 2,000,000 3,000,000 4,000,000 5,000,000 ] 1 2 3 4 5 10 10 10 10 100 = ?
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ENGR 101, Section 100 M. Wellman 5 Dot Product Applies to two equal-sized column vectors: a 1 a 2 a n – 1 a n b 1 b 2 b n – 1 b n = a i i = 1 n b i a b = a T b Multiply a Matrix and a Column Vector Number of columns in the matrix must equal number of elements in the vector. Multiply each matrix row by the column vector. Result is a column vector, one value for each matrix row. A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 A 41 A 42 A 43 = A 11 b 1 + A 12 b 2 + A 13 b 3 A 21 b 1 + A 22 b 2 + A 23 b 3 A 31 b 1 + A 32 b 2 + A 33 b 3 A 41 b 1 + A 42 b 2 + A 43 b 3 b 1 b 2 b 3
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ENGR 101, Section 100 M. Wellman 6 Example 4 0 –1 3 7 0 0 1 0 1 0 1 3 2 1 = 4 × 3 + 0 × 2 + (–1) × 1 3 × 3 + 7 × 2 × 1 0 × 3 + 1 × 2 × 1 1 × 3 + 0 × 2 + 1 × 1 Example 4 0 –1 3 7 0 0 1 0 1 0 1 3 2 1 = 12 + 0 – 1 9 + 14 + 0 0 + 2 + 0 3 + 0 + 1 = 11 23 2 4
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ENGR 101, Section 100 M. Wellman 7 Exercise 1 2 1 0
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This note was uploaded on 05/25/2011 for the course ENGR 101 taught by Professor Ringenberg during the Fall '07 term at University of Michigan.

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15. Matrix Calculation - ENGR 101, Section 100 Matrix...

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