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Unformatted text preview: Economics 146, Fall 2007 Preliminaries Review of Linear Algebra. In this course all numbers are real . When using vectors and matrices, real numbers are often called scalars . A vector x is a finite, ordered set of real numbers x 1 , x 2 , . . . , x n . There are a couple of other ways to say this this. One is to call x an n-tuple of numbers. Another way to say this is that x is a real-valued function ( x i ) defined on a set of integers i = 1 , . . . , n . All three of these phrases mean the same thing (and you may have seen them in previous courses). The numbers x i are called the components of x . By convention, in this course vectors are written as column vectors. To write a row vector, we use the transpose symbol. Thus, x = 3 12- 4 is a column vector, while x T = h 3 12- 4 i is a row vector. We add vectors by adding components, and we multiply vectors by scalars by multiplying each component by the scalar. Thus, for vectors x , y , and scalar , we have x + y = x 1 x 2 . . . x n + y 1 y 2 . . . y n = x 1 + y 1 x 2 + y 2 . . . x n + y n x = x 1 x 2 . . . x n = x 1 x 2 . . . x n The set of all vectors with n components is the real, n-dimensional vector space R n . An m n matrix A is a rectangular array of numbers a ij with m rows and n columns. Thus A = " 3 2- 1- 4 0 2 . 1 # is a 2 3 matrix. Note also that a column vector x R n is an n 1 matrix and the corresponding row vector is an 1 n matrix. The transpose of a matrix A = ( a ij ) is A T = ( b ij ) where b ij = a ji . In the example in 1 the previous paragraph the 2 3 matrix A has as its transpose the 3 2 matrix A T = 3- 4 2- 1 2 . 1 There are three algebraic matrix operations: addition, scalar multiplication, and matrix...
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- Fall '07