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Exercise7.4. If possible find a bases of {(x, y, z, t) € R¹ : x+y+z+ t = 0} containing the vectors (i) (1,-1,0,0) and (1,-1, 1, -1) (ii) (1,-1,1,-1) and (-1, 1,-1,1) (iii) (1, 2, 3, -6)

Exercise7.4. If possible find a bases of {(x, y, z, t) € R¹ : x+y+z+ t = 0} containing the vectors (i) (1,-1,0,0) and (1,-1, 1, -1) (ii) (1,-1,1,-1) and (-1, 1,-1,1) (iii) (1, 2, 3, -6)

Advanced Engineering Mathematics
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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Can you explain the solutions to these 3 parts of the question I attached the answers
Exercise7.4. If possible find a bases of {(x, y, z, t) € R¹ : x+y+z+ t = 0} containing the vectors (i) (1,-1,0,0) and (1,-1, 1,-1) (1,-1,1, -1) and (-1,1,-1,1) (ii) (iii) (1, 2, 3, -6)
Transcribed Image Text:Exercise7.4. If possible find a bases of {(x, y, z, t) € R¹ : x+y+z+ t = 0} containing the vectors (i) (1,-1,0,0) and (1,-1, 1,-1) (1,-1,1, -1) and (-1,1,-1,1) (ii) (iii) (1, 2, 3, -6)
Exercise 7.4 V= {(x, y, z,t) = R¹ : x+y+z+t=0} Note that the dimension of V is 3. (i) V1 = (-1,-1, 0, 0) and v₂ = = (1,-1,1, -1) They belong to V and they are linearly independent. Thus it is possible to find v3 such that {v1, v1, v1} is a basis of V. I It is enough to choose v3 E and v3 # {v1, v2v3}, for example v3 = (0, 1,-1,0) (ii) Since the vectors are not linearly independent, it is not possible to complete to a basis. (iii) v₁ = (1, 2, 3, -6) EV thus we can find v2, v3 EV such that (v1, v2, v3} is a basis of V. Note that: V = {(1,-1,0,0), (1, 0, -1,0), (1, 0, 0, -1)} and we can choose v2 = (1,-1, 0, 0), v3 = (1,0,-1,0) Exercise 7.5
Transcribed Image Text:Exercise 7.4 V= {(x, y, z,t) = R¹ : x+y+z+t=0} Note that the dimension of V is 3. (i) V1 = (-1,-1, 0, 0) and v₂ = = (1,-1,1, -1) They belong to V and they are linearly independent. Thus it is possible to find v3 such that {v1, v1, v1} is a basis of V. I It is enough to choose v3 E and v3 # {v1, v2v3}, for example v3 = (0, 1,-1,0) (ii) Since the vectors are not linearly independent, it is not possible to complete to a basis. (iii) v₁ = (1, 2, 3, -6) EV thus we can find v2, v3 EV such that (v1, v2, v3} is a basis of V. Note that: V = {(1,-1,0,0), (1, 0, -1,0), (1, 0, 0, -1)} and we can choose v2 = (1,-1, 0, 0), v3 = (1,0,-1,0) Exercise 7.5
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