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31. Prove parts (a), (c), and (d) of Theorem 3.1.1.

31. Prove parts (a), (c), and (d) of Theorem 3.1.1.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.3: Lines And Planes
Problem 34EQ
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FIGURE Ex-29 30. What is the sum of all radial vectors of a regular n-side gon? (See Figure Ex-29.) Working with Proofs 31. Prove parts (a), (c), and (d) of Theorem 3.1.1. 32. Prove parts (e)-(h) of Theorem 3.1.1. 33. Prove parts (a)-(c) of Theorem 3.1.2.
Transcribed Image Text:FIGURE Ex-29 30. What is the sum of all radial vectors of a regular n-side gon? (See Figure Ex-29.) Working with Proofs 31. Prove parts (a), (c), and (d) of Theorem 3.1.1. 32. Prove parts (e)-(h) of Theorem 3.1.1. 33. Prove parts (a)-(c) of Theorem 3.1.2.
EXAMPLE 3 | Algebraic Operations Using Components If v = (1, -3, 2) and w = (4, 2, 1), then v+w=(5,-1, 3), 2v = (2,-6,4) -w = (-4,-2, -1), v-w=v+(-w) = (-3, -5,1) The following theorem summarizes the most important properties of ve tions. Theorem 3.1.1 If u, v, and w are vectors in R", and if k and m are scalars, then: (a) u + v=v+u (b) (u+v)+w=u+ (v + w) (c) u+0=0+u=u (d) u + (-u) = 0 (e) k(u + v) = ku + kv (f) (k+m)u = ku + mu (g) k(mu) = (km)u (h) lu= u We will prove part (b) and leave some of the other proofs as exercises. Proof (b) Let u = (U₁, U₂, ..., Un), V = (U₁, U₂, ..., Un), and w = (W1, W2,...,U (u + v) + w = ((U₁, U₂,..., Un) + (V₁, V₂,..., Un)) + (W₁, W₂,..., wn) [Vect = = (U₁ + V₁, U₂+U2,..., Un + Un) + (W₁, W2,..., wn) = ((U₁ + v₁) + w₁, (u₂ + V₂) + W2,..., .., (un + vn) + wn) [Vect [Regr = (u₁ + (v₁ + w₁), U₂ + (V₂ + W₂),..., Un + (Un+wn))
Transcribed Image Text:EXAMPLE 3 | Algebraic Operations Using Components If v = (1, -3, 2) and w = (4, 2, 1), then v+w=(5,-1, 3), 2v = (2,-6,4) -w = (-4,-2, -1), v-w=v+(-w) = (-3, -5,1) The following theorem summarizes the most important properties of ve tions. Theorem 3.1.1 If u, v, and w are vectors in R", and if k and m are scalars, then: (a) u + v=v+u (b) (u+v)+w=u+ (v + w) (c) u+0=0+u=u (d) u + (-u) = 0 (e) k(u + v) = ku + kv (f) (k+m)u = ku + mu (g) k(mu) = (km)u (h) lu= u We will prove part (b) and leave some of the other proofs as exercises. Proof (b) Let u = (U₁, U₂, ..., Un), V = (U₁, U₂, ..., Un), and w = (W1, W2,...,U (u + v) + w = ((U₁, U₂,..., Un) + (V₁, V₂,..., Un)) + (W₁, W₂,..., wn) [Vect = = (U₁ + V₁, U₂+U2,..., Un + Un) + (W₁, W2,..., wn) = ((U₁ + v₁) + w₁, (u₂ + V₂) + W2,..., .., (un + vn) + wn) [Vect [Regr = (u₁ + (v₁ + w₁), U₂ + (V₂ + W₂),..., Un + (Un+wn))
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