**Terminology:**- column vector, scalar, zero vector, linear combination, span, matrix, matrix equation, matrix-vector product, identity matrix.
**Objectives:**- learn about column vectors and their algebraic properties; understand what a linear combination is; understand what the span of a set of vectors is; learn how to multiply a vector by a matrix; understand the connections between matrix-vector equations and linear systems; understand the algebraic properties of matrix-vector products.
**Reading Assignment:**- Chapter 1, Sections 1.3-1.4 (pages 27-46).
**Lesson Outline****Key Ideas and Discussion:**-
A (column) vector in
*n*dimensions (in ) is a column of*n*real numbers. Vectors can be multiplied by scalars, and added together. The algebraic properties for vectors are similar to the properties satisfied by scalars. A linear combination of vectors is a sum of scalar multiples of the vectors. A linear system is just a vector equation where the left-hand side of the equation is a linear combination of some vectors with the unknowns for the system used as scalar multipliers. The right-hand side of the equation is a vector of right-hand sides for the linear system. The set of all possible linear combinations of a set of vectors is called the span of the set. In two (three) dimensions the span can interpreted geometrically as a line (plane). In general, the span of a set of*n*-vectors is a subset of ; sometimes the span includes all of and sometimes it does not.The result of multiplying an

*n*-vector by an matrix*A*is an*m*-vector () that is a linear combination (with the 's as multipliers) of the columns of*A*. The matrix equation is another way of writing down a linear system whose coefficient matrix is*A*and right-hand side is . When we try to solve a linear system, we are trying to find scalars so that if we use these scalars as multipliers for the columns of*A*and add the results together, then we get . Another way to consider this process is to say that we are trying to determine if is in the span of the columns of*A*. The columns of*A*span if and only if has a solution for all possible 's in . **Practice Problems:**-
1.3.3, 5, 9,
11,
19, 25 (pp. 36-37);
1.4.3, 5, 9, 15, 27, 29 (pp. 46-47).
**Assignment**-

Fri Jul 24 14:13:29 PDT 1998