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).

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).

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 ??

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.

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?

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