1. The Matrix Equation A x = b (9/16/05) Definition. If A is an m  n matrix with columns a 1, a 2,…, a n and x is a vector in R n, then the product of A and x, denoted A x, is the linear combination of the columns of A using the corresponding entries of x as the weights.

2. Three ways to view things: Theorem. If A is an m  n matrix with columns a 1, a 2,…, a n and b is a vector in R m, the matrix equation A x = b has the same solution set as the vector equation x 1 a 1 + x 2 a 2 +…+ x n a n = b, which in turn has the same solution set as the system of linear equations with augmented matrix [a 1 a 2 … a n b]

3. Existence of Solutions Theorem. If A is an m  n matrix. Then the following statements are equivalent (i.e., for a given A, either they are all true, or they are all false): For each b in R m, the equation A x = b has a solution x. Each b in R m is a linear combination of the columns of A. The columns of A span R m. A has a pivot position in every row.

4. The row-vector rule for computing A x Though the definition of the product of an m  n matrix A and a vector x of length n is made in terms of scalars and vectors (see first slide), the computation is simply that each row of the answer is the sum of the products of the entries of that row of A and the entries of x. Check this idea by an example….

5. Arithmetic Properties It is true, and easy to check by some examples, that if A is an m  n matrix, u and v are vectors in R n, and c is a scalar, then: A (u + v) = A u + A v, and A (c u) = c (A u).

6. Assignment On Monday, we will meet in the MCS lab (Harder 209) and do some machine computations of the row reduction algorithm. Review that algorithm again in preparation. For next Wednesday, please: Read Section 1.4. Do the Practice and Exercises 1-19 odd and 23.