1. Vectors and Vector Equations (9/14/05) A vector (for us, for now) is a list of real numbers, usually written vertically as a column. Geometrically, it’s an arrow. The set of all vectors with two entries is called R 2 (“R-two”), and is geometrically just the real plane. The set of all vectors with n entries is called R n, geometrically real n -space (e.g., R 3 is real 3-space).

2. Vector Arithmetic: Addition and Scalar Multiplication Two vectors can be added “component- wise”, and a vector can be multiplied by a number (called a scalar) by simply multiplying each entry. The usual rules of arithmetic (commutativity and associativity of addition, the distributive law, etc.) hold for vectors and scalars (see page 32).

3. Linear Combinations Given vectors v 1, v 2,…, v n and scalars c 1, c 2,…, c n, the vector y = c 1 v 1 + c 2 v 2 + … + c n v n is called a linear combination of v 1, v 2,…, v n. The numbers c 1, c 2,…, c n are called the weights. Given any vector y, how can we figure out if it’s a linear combination of v 1, v 2,…, v n ??

4. Vectors Equations and Linear Systems Fact: Given the vectors a 1, a 2,…, a n, and b, solving the vector equation x 1 a 1 + x 2 a 2 +…+ x n a n = b is exactly the same solving the linear system associated with the augmented matrix [a 1 a 2 … a n b] ! The number of variables in the system is (obviously) n, the number of vectors; the number of equations is the length of the vectors.

5. The Span of a Set of Vectors If v 1, v 2,…, v n are vectors in R n, the set of all linear combinations of v 1, v 2,…, v n is called the subset of R n spanned or generated by v 1, v 2,…, v n, and is denoted Span(v 1, v 2,…, v n ). For example, in R 3, two vectors which are not scalar multiples will span….what?

6. Assignment for Friday Read Section 1.3 In that section, do the Practice and do Exercises 1 – 15 odd, 19, 21, 23, and 29.