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Vectors and Vector Equations (9/14/05)

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.

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Vectors and Vector Equations (9/14/05)

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  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 R2 (“R-two”), and is geometrically just the real plane. • The set of all vectors with n entries is called Rn , geometrically real n -space (e.g., R3 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 v1, v2,…, vn and scalarsc1, c2,…, cn, thevector y = c1 v1 + c2 v2 + … + cn vnis called a linear combination ofv1, v2,…, vn . The numbers c1, c2,…, cnare called the weights. • Given any vector y, how can we figure out if it’s a linear combination of v1, v2,…, vn ??

  4. Vectors Equations and Linear Systems • Fact: Given the vectors a1, a2,…, an , and b, solving the vector equationx1a1 + x2a2 +…+ xnan = bis exactly the same solving the linear system associated with the augmented matrix [a1a2 … anb] ! • 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 v1, v2,…, vn are vectors in Rn, the set of all linear combinations of v1, v2,…, vn is called the subset of Rnspanned or generated by v1, v2,…, vn , and is denoted Span(v1, v2,…, vn). • For example, in R3, 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.

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