Fundamentals of linear algebra (The Intext series in by Dennis B. Ames

By Dennis B. Ames

Ebook by way of Ames, Dennis Burley

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If k is a real number, what is JkJ? 3 is just an algebraic formulation of something we have noticed in R2 • The length of (an arrow for) 2v is twice the length of v; the length of -tv is one half the length of v, and so on. 4 Definition. A unit vector is a vector oflength 1. 5 Examples. R~ill • • Each of the standard basis vectors in Rn is a unit vector. Wt ffi= [~] ~ ~coors ~ ril· ~ rrl· · · •• ~ r~ · me e1 2 is a umt vecoor m R e, an+~] is a umt vector m R'. 3 to make unit vectors. If vis any vector (except the zero 1 1 1 vector) and we put k = M' then llkvll = lklllvll = M llvll = 1.

Xn = Yn· Chapter 1. Euclidean n-Space 26 45. Suppose u and v are vectors and u is a scalar multiple of v. Need v be a scalar multiple of u? Explain. 46. Let u, v and w be vectors. Show that any linear combination of u and v is also a linear combination of u, v and w. 47. Suppose u, v and ware vectors and w = 2u - 3v. Show that any linear combination of u, v and w is actually a linear combination of just u and v. 48. (a) [BB] Explain how one could use vectors to show that three points X, Y and Z are collinear.

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