16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-3A T =⎡⎢⎣2 1 3−1 2 −23 1 2C = A · B =⎤⎡⎢⎣−1−75⎤⎥⎦⎥⎦ , (A.B) T = [ 1 −3 −2 ]Fact. (A · B) T = B T · A T .Let e i be the n−vector with zeroes everywhere except in the i−th positionand a 1 there. This is called the standard i−th unit n−vector.The n × n matrix whose i−th row consists of a 1 in the i−th positionand zeroes elsewhere is called the n × n identity matrix, and is denoted I n(or simply I if the context makes the size clear).For any m × n matrix A we haveI m A = AI n = A.2 Multiplication of matrices by row and columnvectorsLet p and n be positive integers.Let u 1 , u 2 , . . . , u n be n vectors in R p , and let a 1 , a 2 , . . . , a n be n realnumbers.The expressionu = a 1 u 1 + a 2 u 2 + . . . + a n u nis called the linear combination of the vectors{u 1 , u 2 , . . . , u n }with coefficients {a 1 , a 2 , . . . , a n }.Any expression of the above form is called a linear combination of vectorsin R p .It is useful to note the following properties of matrix multiplication.