Vector Spaces 2.3 2002/07/31 2003/07/09 14:05:19.358 GMT-5 Michael Haag mjhaag@rice.edu Steven Cox cox@rice.edu Justin Romberg jrom@rice.edu Justin Romberg jrom@rice.edu Richard G. Baraniuk richb@rice.edu Michael Haag mjhaag@rice.edu Mariyah Poonawala mariyah@rice.edu Prashant Singh prash@ece.rice.edu vector vector space vector spaces real vector spaces complex vector spaces linear function spaces This module will define what a vector space is and provide useful examples to the reader.

Introduction Vector space A linear vector space S is a collection of "vectors" such that (1) if f 1 S α f 1 S for all scalars α (where α or α ) and (2) if f 1 S , f 2 S , then f 1 f 2 S To define an abstract linear vector space, we need A set of things called "vectors" (X ) A set of things called "scalars" (A ) A vector addition operator ( ) A scalar multiplication operator (* ) The operators need to have all the properties of given below. Closure is usually the most important to show.

Vector Spaces If the scalars α are real, S is called a real vector space. If the scalars α are complex, S is called a complex vector space. If the "vectors" in S are functions of a continuous variable, we sometimes call S a linear function space

Properties We define a set V to be a vector space if x y y x for each x and y in V x y z x y z for each x , y , and z in V There is a unique "zero vector" such that x 0 x for each x in V For each x in V there is a unique vector x such that x x 0 . 1 x x ( c 1 ⁢ c 2 ) x c 1 ( c 2 ⁢ x ) for each x in V and c 1 and c 2 in ℂ . c x y c x c y for each x and y in V and c in ℂ . c 1 c 2 x c 1 x c 2 x for each x in V and c 1 and c 2 in ℂ .

Examples n real vector space n complex vector space L 1 f t t f t f t is a vector space L ∞ f t f ⁡ ( t )  is bounded f t is a vector space L 2 f t t f t 2 f t finite energy signals is a vector space L 2 0 T finite energy functions on interval [0,T] ℓ 1 , ℓ 2 , ℓ ∞ are vector spaces The collection of functions piecewise constant between the integers is a vector space

ℝ + 2 x 0 x 1 x 0 0 x 1 0 x 0 x 1 is not a vector space. 1 1 ℝ + 2 , but α α 0 α 1 1 ℝ + 2 D z z 1 z is not a vector space. z 1 1 D , z 2 D , but z 1 z 2 D , z 1 z 2 2 1 Vector spaces can be collections of functions, collections of sequences, as well as collections of traditional vectors (i.e. finite lists of numbers)