MATH 415 Study Guide - Final Guide: Khan Academy, Vector Space, Euclidean Vector

• School: University of Illinois
• Department: Mathematics
• Course: MATH 415
• Professor: J Li

Document Summary

Suggested practice exercises: chapter 2. 1: 1, 2, 10, 11, 17, 18. The objects of such a set are called vectors. A vector space is a non-empty set v of objects, called vectors, for which linear combinations make sense. More precisely: on v there are de ned two operations, called addition and multiplication by scalars (real numbers), subject to the ten axioms below. The axioms must hold for all u, v, and w in. 1: cu is in v . (v is closed under scalar multiplication . , c(u + v) = cu + cv, (c + d)u = cu + du, (cd)u = c(du), 1u = u. Let m2x2 = (cid:26)(cid:20)a b c d(cid:21) : a, b, c, d r(cid:27) . We need to say what the two operations are. Addition: (cid:20)a b d(cid:21) +(cid:20)e g h(cid:21) = (cid:20)a + e c + g f c. Scalar multiplication: e (cid:20)a b d(cid:21) = (cid:20)ea eb ed(cid:21) . ec c.