1. Dimension of a Vector Space (11/9/05) Theorem. If the vector space V has a basis consisting of n vectors, then any set of more than n vectors in V must be linearly dependent. (Proof? Use coordinate systems.) Hence if V has a basis B 1 which has n vectors and if B 2 is another basis for V, then B 2 must also has n vectors in it.

2. Definition of Dimension If a vector space V is spanned by a finite set of vectors, then V is said to be finite-dimensional, and its dimension is the (unique) number of elements in any basis of V. Otherwise V is said to be infinite-dimensional. Dimension of V is denoted dim(V ).

3. Examples dim(R n ) = (duh!) dim(P n ) = ? Let M m,n be the set of all m by n matrices. M m,n is a vector space (why?). What is a natural basis for M m,n ? What is dim(M m,n )? Let P be the set of all polynomials in a variable t. Show that P is infinite- dimensional.

4. Subspaces If H is a subspace of a finite- dimensional vector space V, then dim(H )  dim(V) Example: Subspaces of R 3 : 0-dimensional: the origin 1-dimensional: lines through the origin 2-dimensional: planes through the origin 3-dimensional: all of R 3

5. Null and Column Spaces If A be an m by n matrix (hence representing a linear transformation from R n to R m, then dim(Nul(A)) = the number of free variables in A. dim(Col(A)) = the number of pivot columns of A. Hence dim(Nul(A)) + dim(Col(A)) = n.

6. Assignment for Friday Read Section 4.5. Do Exercises 1, 3, 5, 9, 11, 13, 15, 17, 19, 21, and 23.