Inner products, orthogonality, and orthonormal bases from a college linear algebra course.
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- Definition of inner product
- A function on two vectors returning a scalar with properties: linearity, symmetry, and positive definiteness
- Standard inner product in R^n
- u dot v = u_1*v_1 + u_2*v_2 + ... + u_n*v_n
- What does bilinearity of an inner product mean?
- It is linear in each argument: <au+bv, w> = a<u,w> + b<v,w> and similarly in the second slot
- Symmetry property of inner product
- <u,v> = <v,u>
- Positive definiteness of inner product
- <v,v> > 0 when v is nonzero, and <v,v> = 0 if and only if v = 0
- When are two vectors orthogonal?
- When their inner product is zero: <u,v> = 0
- Is the zero vector orthogonal to other vectors?
- Yes, the zero vector is orthogonal to every vector
- What is an orthonormal set?
- A set where vectors are pairwise orthogonal and each has unit norm (length 1)
- What is an orthonormal basis?
- A basis consisting of vectors that are pairwise orthogonal and each has unit norm
- How do you compute the norm of a vector v?
- ||v|| = sqrt(<v,v>)
- Distance between vectors u and v
- d(u,v) = ||u - v|| = sqrt(<u-v, u-v>)
- Formula for the angle theta between nonzero vectors u and v
- cos(theta) = <u,v> / (||u|| * ||v||)
- First step of the Gram-Schmidt process
- Take the first vector v_1 and normalize it: e_1 = v_1 / ||v_1||
- General step of the Gram-Schmidt process
- For vector v_i, subtract its projections onto all previous orthonormal vectors, then normalize the result
- Orthogonal projection of u onto vector v
- proj_v(u) = (<u,v> / <v,v>) * v
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