s02lec02c

# s02lec02c - 15.053 Review of Linear Algebra February 7 2002...

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1 15.053 February 7, 2002 z A brief review of Linear Algebra z Linear Programming Models Handouts: Lecture Notes 2 Review of Linear Algebra z Some elementary facts about vectors and matrices. z The Gauss-Jordan method for solving systems of equations. z Bases and basic solutions and pivoting. 3 Elementary Facts about Vectors [ ] 1234 vvvvv = is called a row vector . 1 2 3 4   =  t v v v v v The transpose of v is a column vector . [ ] wwwww = is another row vector. The inner product of vectors w and v is given by: 11 22 33 44 v w vw vw vw =+ ++ D 4 Matrix Multiplication () = ij A a = ij Bb == × ij Cc AB 1 = = k kj n ik j i ab c Suppose that A has n columns and B has n rows. 5 Multiplying Matrices Let C = ( c ij ) = AB. Then c ij is the inner product of row i of A and column j of B . 21 22 23 11 12 13 31 32 33 aaa A = 13 23 11 12 21 22 31 33 32 b b b bb Bbb = For example, what is c 23 ? 23 21 13 22 23 23 33 ca b a ba b =++ 6 Multiplying Matrices Let C = ( c ij ) = A × B. Then each column of C is obtained by adding multiples of columns of A. 1 2 3 100 123 456 1 0 4 5 6 7 8 9 1 789 ×= 12 3 100 4 10 5 6 78 9   +  Similarly, each row of C is obtained by adding multiples of rows of B.

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Elementary Facts about Solving Equations 12 3 4 0 21 16 xx x    ++ =       Find a linear combination of the columns of A that equals b . Solve for Ax = b , where 2 × 1 12 4 21 1 A = 1 2 3 x x x x = 0 6 b   =     2 × 3 3 × 1 123 240 26 xxx ++= +−= 8 Solving a System of Equations To solve a system of equations, use Gauss-Jordan elimination. x1 x2 x3 x4 1241 = 0 21- 1 - 1 = 6 -1 1 2 2 = -3 9 The system of equations 1 2 -1 2 1 1 4 -1 2 1 -1 2 = = = 0 6 -3 x 1 x 2 x 3 x 4 10 Pivot on the element in row 1 column 1 = = = 0 0 -3 3 -9 6 -3 3 1 2 4 1 0 6 -3 Subtract 2 times constraint 1 from constraint 2.
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s02lec02c - 15.053 Review of Linear Algebra February 7 2002...

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