By Stephen Andrilli; David Hecker
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Example text
Xn ] for some real numbers x1 , . . , xn . Now, x · x ϭ x12 ϩ · · · ϩ xn2 , by definition of the dot product. However, x ϭ x12 ϩ · · · ϩ xn2 , by definition of the length of a vector. Therefore, x both sides are equal to 2 ϭ x · x, because x12 ϩ · · · ϩ xn2 . The paragraph form should contain the same information as the step-by-step form and be presented in such a way that a corresponding step-by-step proof occurs naturally to the reader. We present most proofs in this book in paragraph style.
4 Scalar multiples of x ϭ [4, Ϫ5] (all vectors drawn with initial point at origin) 6 CHAPTER 1 Vectors and Matrices the vector 2x points in the same direction as x but is twice as long. The vectors Ϫ3x and Ϫ 12 x indicate movements in the direction opposite to x, with Ϫ3x being three times as long as x and Ϫ 12 x being half as long. In general, in Rn , multiplication by c dilates (expands) the length of the vector when |c| > 1 and contracts (shrinks) the length when |c| < 1. Scalar multiplication by 1 or Ϫ1 does not affect the length.
9. Result 4 Let x and y be nonzero vectors in Rn . Then x · y ϭ x is a positive scalar multiple of x. y if and only if y In an “if and only if” proof, it is usually good to begin by stating the two halves of the “if and only if” statement. This gives a clearer picture of what is given and what must be proved in each half. In Result 4, the two halves are 1. Suppose that y ϭ cx for some positive c ∈ R. Prove that x · y ϭ x 2. Suppose that x · y ϭ x y . y . Prove that there is some c > 0 such that y ϭ cx.