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For a = (a1, a2, a3) € R³ and b = (b₁,b2, b3) = R³, define: (a, b) = (a₁ + a₂) b₁ + (a₁ + 3a2) b2 + 3a3b3. (a) Prove that this defines an inner product on R³. (b) Find an orthonormal basis of R³ with respect to this inner product.

For a = (a1, a2, a3) € R³ and b = (b₁,b2, b3) = R³, define: (a, b) = (a₁ + a₂) b₁ + (a₁ + 3a2) b2 + 3a3b3. (a) Prove that this defines an inner product on R³. (b) Find an orthonormal basis of R³ with respect to this inner product.

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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For a = (a1, a2, a3) € R³ and b = (b₁,b2, b3) € R³, define: (a, b) = (a₁ + a2) b₁ + (a₁ + 3a2)b2 + 3a3b3. (a) Prove that this defines an inner product on R³. 3 (b) Find an orthonormal basis of R³ with respect to this inner product.
Transcribed Image Text:For a = (a1, a2, a3) € R³ and b = (b₁,b2, b3) € R³, define: (a, b) = (a₁ + a2) b₁ + (a₁ + 3a2)b2 + 3a3b3. (a) Prove that this defines an inner product on R³. 3 (b) Find an orthonormal basis of R³ with respect to this inner product.
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