Problem 6. Prove that the vectors v₁ = (1,0, -1), U₂ = (1, 2, 1), V3 = (0, -3,2) form a basis for R³. Express each of the standard basis vectors, (1, 0, 0), (0, 1, 0), and (0,0,1) as a linear combination of U₁, U₂, U3.
Problem 6. Prove that the vectors v₁ = (1,0, -1), U₂ = (1, 2, 1), V3 = (0, -3,2) form a basis for R³. Express each of the standard basis vectors, (1, 0, 0), (0, 1, 0), and (0,0,1) as a linear combination of U₁, U₂, U3.
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 \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac744b86-fb77-4dc8-9b17-1f74c21e67b7%2Fec812828-96e7-4407-9c26-fd23a629088d%2Fl4fnuqk_processed.png&w=3840&q=75)
Transcribed Image Text:**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 , where , are called the components of the vectors. In the first part of the problem, we have to prove that form a basis for . In the second part of the problem, we have to express each of the standard basis vectors and as a linear combination of .
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