{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture28

# lecture28 - IE170 Algorithms in Systems Engineering Lecture...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 non-trivial 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 non-trivial 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 )....
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

lecture28 - IE170 Algorithms in Systems Engineering Lecture...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online