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Unformatted text preview: Linear Algebra Basics A vector space (or, linear space) is an algebraic structure that often provides a home for solutions of mathematical models. Linear algebra is a study of vector spaces and a class of special functions (called linear transformations) between them. The terms vector space and linear space are interchangeable. Well use them both. 1. Linear Spaces : The elements of a linear space are called vectors. We can scale and add vectors. By scaling and adding vectors we can build new vectors. This simple construction paradigm is remarkably useful. Definition : A linear space (or vector space ) is a set L together with field of scalars F and two functions (operations); addition + : L L L and (scalar) multiplication : F L L satisfying conditions (a)-(h) below. For us, the scalar field F will almost always be the real numbers R (rarely, for us, it may be the complex numbers C ). Also, for brevity (clarity?), if F and x L and we want to scale x using , then well write x rather than x . (a) x + y = y + x for x , y in L (b) x + ( y + z ) = ( x + y ) + z for x,y,z L (c) there is an element O L such that for each x L , x + O = x (d) for each x L there corresponds y L such that x + y = O (e) for each , F and x L , ( x ) = ( ) x (f) for each , F and x L , ( + ) x = x + x (g) for each F and x,y L , ( x + y ) = x + y , and (h) 1 x = x for each x L . Remarks : If x + y = x + z then y = z . This cancellation law holds because y = O + y = (- x + x ) + y =- x + ( x + y ) =- x + ( x + z ) = (- x + x ) + z = O + z = z . x + x = (0 + 1) x = x so 0 x = O . The zero element O of (b) and y , the additive inverse of x , in (c) are unique; we usually write- x for the additive inverse of x . Since- 1 x + x =- 1 x + 1 x = (- 1 + 1) x = 0 x = O then- x =- 1 x . For us, the scalar field F will always be either the real numbers R or complex numbers C . There are other fields however, for example the rational numbers Q . There are finite fields, e.g., Z p (the integers mod p ) where p is a prime number. Vector spaces over finite fields find applications in, for example, crytography. Well not consider them in this course. Examples of linear spaces: (a) R n : A vector x R n is an n-tuple of real numbers, x = ( x 1 ,x 2 ,...,x n ). The set of real numbers R is the scalar field. Vector additon and scalar multiplication are defined component-wise: if x and y are vectors and R then 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 ) and x = ( x 1 ,x 2 ,...,x n ) := ( x 1 ,x 2 ,...,x n )....
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This document was uploaded on 09/22/2011.
- Spring '09