Inner Products

Question 1

What is an inner product?

Answer 1

A map from (v, w) -> satisfying certain properties.

Question 2

Define linearity in the first variable

Answer 2

The first variable respects addition and scalar multiplication rules

Question 3

Define inner product conjugate transpose

Answer 3

= complex conjugate()

Question 4

Define inner product positivity

Answer 4

is positive (greater than or equal to 0) for all v

Question 5

Define inner product definiteness

Answer 5

= 0 if and only if v = 0

Question 6

What conditions do we use integration for the inner product?

Answer 6

If the vector space V is on a field containing polynomials or functions (R[X], p(x), C([0, 1]))

Question 7

Define the norm of a vector

Answer 7

The norm (or length) of a vector is defined 
||v|| = sqrt()

Question 8

When are two vectors orthogonal?

Answer 8

Two vectors are orthogonal if = 0. Due to symmetry = 0 too

Question 9

Which vector is orthogonal to every vector?

Answer 9

The 0 vector is orthogonal to every vector

Question 10

What is the Pythagorean Theorem?

Answer 10

If V and W are orthogonal vectors

||v+w||^2 = ||v||^2 + ||w||^2

Question 11

What is orthogonal decomposition?

Answer 11

The orthogonal decomposition of vector v in u:

u = (Xv + (u - Xv)) where generally X = {u, v}/(||v||^2)

Question 12

What is the Cauchy-Schwartz Theorem?

Answer 12

|{u, v}| <= ||u|| * ||v||

Question 13

What is the Triangle Inequality?

Answer 13

||u + v|| <= ||u|| + ||v||

Question 14

What is the Parallelogram Inequality?

Answer 14

||u + v||^2 + ||u - v||^2 = 2(||u||^2 + ||v||^2)

Question 15

What is orthonormal?

Answer 15

A set of vectors are orthonormal if all vectors in the set are pairwise orthogonal and have norm 1

Question 16

What can we say about every orthonormal set?

Answer 16

Every orthonormal set of vectors are linearly independent

Question 17

What is an orthonormal basis in V?

Answer 17

An orthonormal set of vectors which are also a basis in V

Question 18

What is the Gram-Schmidt Theorem?

Answer 18

If there is a linearly independent set in V, then an orthonormal set exists such that span{V} = span{E}

Question 19

Can we convert an orthonormal set to an orthonormal basis?

Answer 19

Yes, every orthonormal set can be extended to an orthonormal basis

Question 20

What is the Orthogonal complement?

Answer 20

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^⊥

Question 21

What do we know about U^⊥?

Answer 21

It is a subspace, and if W was a subspace of U, U^⊥ is a subspace of W^⊥

Question 22

Define the direct sum

Answer 22

V = U ⊕ W, where U and W share no elements (the intersection of U and W = {0}).

Question 23

How does the direct sum relate to the orthogonal complement?

Answer 23

If U is a subspace of V, V = U ⊕ U^⊥

Question 24

What is an orthogonal matrix?

Answer 24

An invertible matrix whose column vectors form an orthonormal basis

Question 25

Define linearity in the second variable

Answer 25

The second variable respects addition but

= conjugate(a)