Practice question · Select all that apply
Select every TRUE statement about inner products and orthogonality.
Hints
- One option confuses a condition on angles with a condition on lengths.
- The inner product of the zero vector with anything is 0.
Show the answer
- A. Two vectors are orthogonal exactly when their inner product is 0
- B. The zero vector is orthogonal to every vector
- C. A set of mutually orthogonal nonzero vectors is linearly independent
- D. In an orthonormal basis, the i-th coordinate of v is the inner product of v with the i-th basis vector
Why
Orthogonality of nonzero vectors forces independence, and in an orthonormal basis coordinates need no system solving at all. But an orthogonal set becomes orthonormal only after each vector is divided by its own norm, that normalising step is exactly what Gram-Schmidt finishes with.
Practise Inner Product Spaces and Orthogonality
The app has 6 more questions on this lesson, and keeps your place in the course. Mathematics I is free to start.
More questions on Inner Product Spaces and Orthogonality
- Order the steps of the Gram-Schmidt process applied to a basis v1, v2, v3.
- Gram-Schmidt takes a perfectly good basis and replaces it with an orthonormal one. Why is that worth the…
- Sort each pair of vectors by whether the two are orthogonal.
- A search engine measures document similarity by the angle between vectors rather than the distance. Why is…