MATH115 Study Guide - Scalar Multiplication, Orthogonality, Dot Product

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Vectors have a magnitude and a direction, are denoted by vector arrows, and are said to be in rn. You can perform two main operations with vectors: vector addition, which is the algebraic tail to tip addition of vectors, and scalar multiplication, which is t(cid:126)v where t r and (cid:126)v rn. The linear combinatio of a set of vectors ( (cid:126)v0, . (cid:126)vn is any vector which can be obtained from these vectors through vector addition and scalar multiplication. It has the form a0 (cid:126)v0 + + an (cid:126)vn where a0, . an r (cid:20)1 (cid:21) Example: for (cid:126)a = and (cid:126)b = (cid:20) 3 2 (cid:21) Any vector in r2 is a linear combination of this set. In rn, the dot product of (cid:126)u = (cid:20)3 (cid:21) Is a scalar de ned to be bn an (cid:126)a (cid:126)b = a0b0 + + anbn (cid:126)u (cid:126)v = 3 2 + 5 6 = 36.

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