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Let 01, 02, 03 be three nonzero vectors in a vector space V. Suppose that V1, V2, V3 are orthogonal relative to the (non-degenerate and positive definite) scalar product (): V x V → R, i.e., (V1, V₂) = 0, (v1,v3) = 0, (V2, V3) = 0. (1) Show that 01, 02, 03 are necessarily linearly independent.

Let 01, 02, 03 be three nonzero vectors in a vector space V. Suppose that V1, V2, V3 are orthogonal relative to the (non-degenerate and positive definite) scalar product (): V x V → R, i.e., (V1, V₂) = 0, (v1,v3) = 0, (V2, V3) = 0. (1) Show that 01, 02, 03 are necessarily linearly independent.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.4: Spanning Sets And Linear Independence
Problem 74E: Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors...
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Let V1, V2, V3 be three nonzero vectors in a vector space V. Suppose that V1, V2, V3 are orthogonal relative to the (non-degenerate and positive definite) scalar product (): V x V → R, i.e., (V1, V₂) = 0, (v1,v3) = 0, (V2, V3) = 0. (1) Show that 01, 02, 03 are necessarily linearly independent.
Transcribed Image Text:Let V1, V2, V3 be three nonzero vectors in a vector space V. Suppose that V1, V2, V3 are orthogonal relative to the (non-degenerate and positive definite) scalar product (): V x V → R, i.e., (V1, V₂) = 0, (v1,v3) = 0, (V2, V3) = 0. (1) Show that 01, 02, 03 are necessarily linearly independent.
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