**Problem 6.** Prove that the vectors\[ \mathbf{v}_1 = (1, 0, -1), \quad \mathbf{v}_2 = (1, 2, 1), \quad \mathbf{v}_3 = (0, -3, 2) \]form a basis for $ \mathbb{R}^3 $. Express each of the standard basis vectors, $ (1, 0, 0) $, $ (0, 1, 0) $, and $ (0, 0, 1) $ as a linear combination of $ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 $.

Answered: Problem 6. Prove that the vectors v₁ =…

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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**Problem 6.** Prove that the vectors

\[ \mathbf{v}_1 = (1, 0, -1), \quad \mathbf{v}_2 = (1, 2, 1), \quad \mathbf{v}_3 = (0, -3, 2) \]

form a basis for $ \mathbb{R}^3 $. Express each of the standard basis vectors, $ (1, 0, 0) $, $ (0, 1, 0) $, and $ (0, 0, 1) $ as a linear combination of $ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 $.

Expert Solution

Step 1

A vector is a quantity that obeys the set of axioms. In general, vectors are represented as $A = (A_{1}, A_{2}, A_{3})$ , where $A_{1}, A_{2}$ , $A_{3}$ are called the components of the vectors. In the first part of the problem, we have to prove that $v_{1} = (1, 0, - 1), v_{2} = (1, 2, 1), v_{3} = (0, - 3, 2)$ form a basis for $ℝ^{3}$ . In the second part of the problem, we have to express each of the standard basis vectors $(1, 0, 0), (0, 1, 0)$ and $(0, 0, 1)$ as a linear combination of $v_{1}, v_{2}, v_{3}$ .