Working directly from the definition of a basis, establish the following state- ments: (i) e₁ - e2, e2 - e3, e3 - ₁ is not a basis for R³. (ii) e₁ - e2, e2 - e3, ₁+ 2+ 3 is a basis for R³. (iii) e₁ - e2, e2 - e3, e3 - e₁, e₁ + 2 + 3 is not a basis for R³. In your answer address the following (without referring to the concept of dimension): • Are the vectors linearly independent? • Do the vectors span R³? If so, give a parameterisation for all ways (x, y, z)¹ € R³ can be expressed as a linear combination of the vectors.
Working directly from the definition of a basis, establish the following state- ments: (i) e₁ - e2, e2 - e3, e3 - ₁ is not a basis for R³. (ii) e₁ - e2, e2 - e3, ₁+ 2+ 3 is a basis for R³. (iii) e₁ - e2, e2 - e3, e3 - e₁, e₁ + 2 + 3 is not a basis for R³. In your answer address the following (without referring to the concept of dimension): • Are the vectors linearly independent? • Do the vectors span R³? If so, give a parameterisation for all ways (x, y, z)¹ € R³ can be expressed as a linear combination of the 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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Transcribed Image Text:Working directly from the definition of a basis, establish the following state-
ments:
(i) e₁ - e2, e2 - e3, 3 – ₁ is not a basis for R³.
(ii) e₁ - e2, e2 - e3, е₁ + ₂ + 3 is a basis for R³.
(iii) e₁ - e2, e2 – €3, €3 – е1, е₁ + 2 + 3 is not a basis for R³.
In your answer address the following (without referring to the concept of
dimension):
• Are the vectors linearly independent?
Do the vectors span R³? If so, give a parameterisation for all ways
(x, y, z)™ – R³ can be expressed as a linear combination of the vectors.
●
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