Practice Inner Product Spaces (27) - Inner Product Spaces - Mathematics (Civil Engineering -1)

Practice: Inner Product Spaces

Lesson 27Inner Product Spaces

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8 questions

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3 cards from this section. Quick, good the night before a test.
Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
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2 questions · about 1 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is an inner product space?
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Think about how it measures angles and lengths.
Question 2.
Define orthogonality in the context of vector spaces.
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Relate this to geometry, where what does it mean to be at a right angle?
3Question 3.
What is the property of the Cauchy-Schwarz inequality?
- |⟨u,v⟩| ≤ ||u|| + ||v||
- |⟨u,v⟩| ≤ ||u||·||v||
- ⟨u,v⟩ = ||u|| ||v||
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Think about relationships in geometry.
4Question 4.
True or False: In an inner product space, if |⟨u,v⟩| is greater than ||u||·||v||, then u and v are linearly independent.
- True
- False
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Recall the definition of linear dependence.
Question 5.
Prove that if two vectors are orthogonal in an inner product space, their linear combination with appropriate coefficients will also remain in the span of those vectors.
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Use the properties of linear combinations.
Question 6.
Using the Gram–Schmidt process, convert the vectors u=(1,1,1) and v=(1,2,3) into orthonormal basis vectors.
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Apply the iterative projection formulas carefully.

4 more questions in this section. Enroll free to practise them all.