Inner Products Flashcards
What is an inner product?
A map from (v, w) -> satisfying certain properties.
Define linearity in the first variable
The first variable respects addition and scalar multiplication rules
Define inner product conjugate transpose
= complex conjugate()
Define inner product positivity
is positive (greater than or equal to 0) for all v
Define inner product definiteness
= 0 if and only if v = 0
What conditions do we use integration for the inner product?
If the vector space V is on a field containing polynomials or functions (R[X], p(x), C([0, 1]))
Define the norm of a vector
The norm (or length) of a vector is defined ||v|| = sqrt()
When are two vectors orthogonal?
Two vectors are orthogonal if = 0. Due to symmetry = 0 too
Which vector is orthogonal to every vector?
The 0 vector is orthogonal to every vector
What is the Pythagorean Theorem?
If V and W are orthogonal vectors
||v+w||^2 = ||v||^2 + ||w||^2
What is orthogonal decomposition?
The orthogonal decomposition of vector v in u:
u = (Xv + (u - Xv)) where generally X = {u, v}/(||v||^2)
What is the Cauchy-Schwartz Theorem?
|{u, v}| <= ||u|| * ||v||
What is the Triangle Inequality?
||u + v|| <= ||u|| + ||v||
What is the Parallelogram Inequality?
||u + v||^2 + ||u - v||^2 = 2(||u||^2 + ||v||^2)
What is orthonormal?
A set of vectors are orthonormal if all vectors in the set are pairwise orthogonal and have norm 1
What can we say about every orthonormal set?
Every orthonormal set of vectors are linearly independent
What is an orthonormal basis in V?
An orthonormal set of vectors which are also a basis in V
What is the Gram-Schmidt Theorem?
If there is a linearly independent set in V, then an orthonormal set exists such that span{V} = span{E}
Can we convert an orthonormal set to an orthonormal basis?
Yes, every orthonormal set can be extended to an orthonormal basis
What is the Orthogonal complement?
For a subspace U of V, the orthogonal complement of U is the set of vectors in V that are orthogonal to every vector in U. Denoted U^⊥
What do we know about U^⊥?
It is a subspace, and if W was a subspace of U, U^⊥ is a subspace of W^⊥
Define the direct sum
V = U ⊕ W, where U and W share no elements (the intersection of U and W = {0}).
How does the direct sum relate to the orthogonal complement?
If U is a subspace of V, V = U ⊕ U^⊥
What is an orthogonal matrix?
An invertible matrix whose column vectors form an orthonormal basis
Define linearity in the second variable
The second variable respects addition but
= conjugate(a)
