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Let V, < ·, · > be inner product vector spaces. Let S C V be some collection of vectors. Let S be the set of all vector in V that are orthongonal to all vectors in S. Show that S- is a subspace. Verify that the following is an inner product on R². < (x1, x2), (Y1, Y2) >= x1Y1 - x1Y2 - x2Y1 + 3x2Y2 Find a vector orthogonal to (1,5) with respect to this new inner product. Find a vector orthogonal to (1,5) with respect to this standard inner product.

Let V, < ·, · > be inner product vector spaces. Let S C V be some collection of vectors. Let S be the set of all vector in V that are orthongonal to all vectors in S. Show that S- is a subspace. Verify that the following is an inner product on R². < (x1, x2), (Y1, Y2) >= x1Y1 - x1Y2 - x2Y1 + 3x2Y2 Find a vector orthogonal to (1,5) with respect to this new inner product. Find a vector orthogonal to (1,5) with respect to this standard inner product.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 12E
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Let V, < ·, · > be inner product vector spaces. Let S C V be some collection of vectors. Let S be the set of all vector in V that are orthongonal to all vectors in S. Show that S- is a subspace.
Transcribed Image Text:Let V, < ·, · > be inner product vector spaces. Let S C V be some collection of vectors. Let S be the set of all vector in V that are orthongonal to all vectors in S. Show that S- is a subspace.
Verify that the following is an inner product on R². < (x1, x2), (Y1, Y2) >= x1Y1 - x1Y2 - x2Y1 + 3x2Y2 Find a vector orthogonal to (1,5) with respect to this new inner product. Find a vector orthogonal to (1,5) with respect to this standard inner product.
Transcribed Image Text:Verify that the following is an inner product on R². < (x1, x2), (Y1, Y2) >= x1Y1 - x1Y2 - x2Y1 + 3x2Y2 Find a vector orthogonal to (1,5) with respect to this new inner product. Find a vector orthogonal to (1,5) with respect to this standard inner product.
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