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Unformatted text preview: IE170: Algorithms in Systems Engineering: Lecture 28 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University April 11, 2007 Jeff Linderoth (Lehigh University) IE170:Lecture 28 Lecture Notes 1 / 12 Linear Algebra Review Another Look at Matrix Multiplication Important Notation If A ∈ R m × n , then A j is the j th column, and a j is the j th row. If A ∈ R m × k ,B ∈ R k × n , then [ AB ] ij = a T i B j . That is, you find the i,j element of the matrix AB , by taking the inner product of the i th row of A with the j th column of B . Naturally this is only defined if A ∈ R m × and B ∈ R × n , wherein also [ AB ] ij = k =1 a ik b kj . Jeff Linderoth (Lehigh University) IE170:Lecture 28 Lecture Notes 2 / 12 Linear Algebra Review Matrix Multiplication: Linear Combinations of Columns Looking at it another way, write B as its columns: B = ( B 1 B 2 ··· B n ) Then the j th column of AB is AB j , or AB = A ( B 1 B 2 ··· B n ) = ( AB 1 AB 2 ··· AB n ) so that each column of AB is a linear combination of the columns of A , and the multipliers for the linear combination are given in the column B j Jeff Linderoth (Lehigh University) IE170:Lecture 28 Lecture Notes 3 / 12 Linear Algebra Review Matrix Multiplication: Linear Combinations of Rows We can also express the relationship in terms of the rows of A A = a 1 a 2 . . . a m ,AB = a 1 B a 2 B . . . a m B So that the i th row of AB is a linear combination of the rows of B , with the weights in the combination coming from the weights in a i . Another Nice Formula ( AB ) T = B T A T . Jeff Linderoth (Lehigh University) IE170:Lecture 28 Lecture Notes 4 / 12 Linear Algebra Review Some Definitions You Already Knew Vectors { A 1 ,A 2 ...A n } are said to be linearly dependent if the zero vector can be written as a nontrivial linear combination of the vectors, or ∃ α 1 ,α 2 ,...,α n not all equal to zero such that n j =1 α j A j = 0 . Alternatively, if A j are columns of A , then the A j are linearly dependent if Az = 0 for some z = 0 Jeff Linderoth (Lehigh University) IE170:Lecture 28 Lecture Notes 5 / 12 Linear Algebra Review More Definitions You Already Knew Vectors { A 1 ,A 2 ...A n } are said to be linearly independent if the zero vector cannot be written as a nontrivial linear combination of the vectors, or n j =1 α j A j = 0 ⇒ α 1 = α 2 = ··· = α n = 0 Alternatively, if A j are columns of A , then the A j are linearly independent if Az = 0 → z = 0 . (0 is the only solution to Az = 0 )....
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 Spring '07
 Ralphs
 Linear Algebra, Vector Space, Systems Engineering, Linear Algebra Review, Jeff Linderoth

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