1. Vector Spaces (10/27/04) The spaces R n which we have been studying are examples of mathematical objects which have addition and scalar multiplication. But there are lots of other such sets: All polynomials of degree  n (called P n ) All functions from D (  R) to R All infinite-tuples of real numbers All m by n matrices, and so on….

2. Abstraction When mathematicians see lots of examples which may look somewhat different but which seem to share some important properties, then they try to abstract, defining a set whose elements are “generic” but which has those important properties. A vector space is such an abstract object.

3. Definition of a Vector Space A vector space V is a non-empty set of elements, called vectors, on which are defined two operations, addition and scalar multiplication. For all vectors u, v, and w and all scalars c and d, the following conditions must hold: u + v is in the vector space V (“closed”) u + v = v + u (“commutative”) (u + v) + w = u + (v + w) (“associative”) There is a zero vector 0 with u + 0 = u

4. Definition Continued Each vector u has a corresponding vector –u such that u + (-u) = 0 The scalar multiple c u of u lies in V (“closed” again) c (u + v) = c u + c v (“distributive”) (c + d ) u = c u + d u (“distributive” again) c (d u) = (c d ) u (“associative” again) 1u = u (“multiplicative identity”)

5. Subspaces A subset H of a vector space V is called a subspace of V if: 0 is in H H is closed under addition, i.e., if u and v are in H, then u + v is also in H H is closed under scalar multiplication, i.e., if u is in H c is a scalar, then c u is also in H Hence a subspace is a vector space in its own right.

6. Examples and Assignment Examples of subspaces: Any line or plane passing through the origin in R 3 is a subspace of R 3 (Why? Check!) The set of polynomials of degree  4 is a subspace of the space of all polynomials of degree  6 (Why? Check!) The set of all continuous functions from R to R is a subspace of the set of all functions from R to R (Why? Check!) Assignment: Read Section 4.1; do Practice and Exercises 1 – 23 odd.