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Unformatted text preview: A QUICK REVIEW OF LINEAR ALGEBRA A Vector: is a vertical array of numbers like: x = 2 3 4 A matrix: two dimensional array of numbers like: A = 2 3 4 3 4 5 Scalar multiplication: 5 x = 10 15 20 and 10 A = 20 30 40 30 40 50 Addition: x + 1 1 1 = 3 4 5 and A + 1 1 1 1 1 1 = 3 4 5 4 5 6 Saigal (U of M) IOE 310 1 / 28 Norm of a vector:  v  = ( ∑ n i =1 x 2 i ) 1 2 is called the norm of v . Inner product of vectors: Let u = u 1 . . . u n and v = v 1 . . . v n then u T v = ∑ n i =1 u i v i where v T is the vector v written as a row, called the transpose of v . CauchySchwartz inequality says that u T v =  u  .  v  . cos( θ v , u ). For matrices A which is m × n and B which is n × m , C = AB the product of the two matrices A and B is a matrix of size m × m . Also C i , j = ∑ n k =1 A i , k B k , j (HOW DOES THIS FORMULA RELATE TO INNER PRODUCT OF VECTORS?). Transpose of a matrix: Given a matrix A that is m × n , its transpose is a n × m matrix A T whose columns are the corresponding rows or A . For example: Saigal (U of M) IOE 310 2 / 28 If A = 2 3 4 3 4 5 , then A T = 2 3 3 4 4 5 What is AA T ? Saigal (U of M) IOE 310 3 / 28 GEOMETRY OF ADDING VECTORS I u v u+v THE GEOMETRY OF ADDING VECTORS x1 x2 u v u+v Angle between u and v Saigal (U of M) IOE 310 4 / 28 USING LINEAR ALGEBRA GaussJordan Elimination Method or the LU factorization Scheme. Linear Independence. Inverse of a Matrix. Saigal (U of M) IOE 310 5 / 28 SYSTEM OF LINEAR EQUATIONS Ax = b where A = a 1 , 1 a 1 , 2 ··· a 1 , n a 2 , 1 a 2 , 2 ··· a 2 , n . . . . . . . . . . . . a m , 1 a m , 2 ··· a m , n , b = b 1 b 2 . . . b m , x = x 1 x 2 . . . x n . Here A is an m × n matrix, b is an m vector and x is an n vector. This system can have a unique solution, no solution and an infinite number of solutions....
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 Spring '08
 Saigal
 Linear Algebra, IOE, nonbasic variables, Saigal

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