Q/prove that if d (u,v) >2 then there isa vertex z in G st. d (u,v) = d(u,z)+d(z₁v)

We need to prove that if the distance between two vertices u and v in a graph G is at least 2, then…

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Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CM: Cumulative Review

Problem 4CM: Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4,...

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Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4, u5 and u6. Then verify your solution. v=(10,30,13,14,7,27) u1=(1,2,3,4,1,2) u2=(1,2,1,1,2,1) u3=(0,2,1,2,1,1) u4=(1,0,3,4,1,2) u5=(1,2,1,1,2,3) u6=(3,2,1,2,3,0)
Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if the matrix [x1y11x2y21x3y31] has rank less than 3.
4. Consider -- -- *-H = = V3-1 2 k a) For what values of k is v3 in Span {V₁, V₂}? Explain. b) For what values of k is {v₁, v2, V3} linearly dependent? Explain. [-51 6 0
2. Triangle ABC with vertices (0, 7), B(7, 3), and C(1, 4): (x, y) → (x – 3, y – 4) A' (_ , B’(__ C' (_ 111
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The plane z = Ax+By+C is to be "fitted" to the following points (XkYkZk): (1,-1,0), (− 1, −1,1), (1,-1,-1), (0,0,1). Find the values of A, B, 4 and C that minimize Σ (AxK+Byk + C-ZK) ², the sum of the squares of the deviations. k=1 A = B = C = (Simplify your answers.)
ez V3 V2 V5 V1 V4 (a) Which edges are incident on v,? (b) Which vertices are adjacent to v? (c) Which edges are adjacent to e,?
Q7) True or False, and Justify your answer. If v1, v2, ... V4 are in R* and v3 = 2v1 + v2, then {v1, V2, V3, V4} is linearly independent.
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A triangle has vertices at R(1, 1), S(-2,-4), and T(-3,-3). The triangle is transformed according to the rule Ro 270. What are the coordinates of S'? O (4, 2) O (2, 4) O (2, 4) O (4,-2)

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Q/prove that if d (u,v) >2 then there is a vertex z in G st. d (u,v) = d(u, z)+d(z₁v)