1. (a)i. Define what it means for vectors v₁,...,vn of a vector space V to be linearlydependent.ii. Show that the set of vectors S = {(2, 1, 3), (2, 3, 4), (4, 4, 5), (2, 5, 6)} spansR³ but is not a basis for R³. Write down a subset of S that does form a basisfor R³.

Vectors in a vector space are linearly dependent if there exist constants (not all zero) such…

Answered: (a) i. Define what it means for vectors…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The prove that the vectors A,B and C form a basis. And decompose the vector D by basis vectors A, B and C. a = (3,1,2); b = (-4,3,-1), c = (2,3,4), d = (14,14,20)
2. Consider the following sets of vectors. Show that each set (i) contains the zero vector, (ii) is closed under addition and scalar multiplication. Then find a basis for each set and give the dimension. (a) W is the set of vectors of the form (3s, -2s, s). (b) W is the set of vectors of the form (t – 2, 0,6 – 3t). (c) H is the set of vectors of the form (2b, a, 3b) (d) H is the set of vectors of the form (-b+ 2c, b, 4c),
1.
2- Show whether the set of vectors S={ ,,} is a basis of R3 ?
5. Determine the courdinate vector of V =(8,-4,-12) velative to the basis S = {C,,1),(1,s,-3), (2,2,113
(3,0,0), v2 = (1,2,0), and v3 = a) Are the vectors linearly independent or linearly dependent? Given vectors v, = (2, –4,5). b) Do the vectors form a basis for R3? Why or why not?
Say you are given two vectors x and y represented using different basis sets {ê;} and {f;}, of the same dimension N. That is, x = y = E y;f;. E xie; and N you are told that a given vector x has the representation x = (1, 1)" in basis {ê;} and the representation x = (-1,1)" in basis {f}}. How might the two basis sets be related? Draw a figure to justify X = your answer.
: Let S = {v1, v2}, where v = standard vectors when added to the set S produces a basis for R? (6,-5, 7,-1) and v2 = (-12, 20, -28, 4). Which of the following combination of (A) v3 = (1, 0, 0, 0), v4 = (0, 0, 1, 0) and vs = (0, 0, 0, 1) (B) v3 = (1, 0, 0, 0), v4 = (0, 1, 0, 0) and vs = (0, 0, 0, 1) (C) v3 = (1, 0, 0, 0) and v4 = (0, 0, 0, 1) (D) v3 = (1, 0, 0, 0), v4 = (0, 1, 0, 0) and vs = (0. 0, 1, 0) (E)v3 = (1, 0, 0, 0) and v4 = (0, 1, 0, 0) (F) v3 = (1, 0, 0, 0) and v4 = (0, 0, 1, 0) (G) v3 = (0, 1, 0, 0) and v4 = (0, 0, 0, 1) %3D %3D (H) v3 = (0, 1, 0, 0), v4 = (0, 0, 1, 0) and vs = (0. 0, 0, 1) %3D
The coordinate vector of the vector (1,2,2) in the basis B = {u= (1,1,1); v = (1,0,1); w= (0,0,1) } is : O A. (1,2,2) O B. (2,-1,1) O C. (2,1,3) O D. (1,2,1)

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(a) i. Define what it means for vectors v₁,..., n of a vector space V to be linearly dependent. ii. Show that the set of vectors S = {(2, 1, 3), (2, 3, 4), (4, 4, 5), (2, 5, 6)} spans R³ but is not a basis for R³. Write down a subset of S that does form a basis for R³.