6) Inner products, Orthogonality, and Isometries

Question 1

What is the Inner Product of two vectors written as a transpose

Answer 1

Question 2

What is the inner product of complex vectors (Hermitian inner product)

Answer 2

Question 3

What is the conjugate transpose

Answer 3

A^∗ of A is the n × m matrix obtained by both transposing A and taking the complex conjugate of each of its entries.

Question 4

What are the properties of the conjugate transpose

Answer 4

Question 5

What are the propeties of the inner product

Answer 5

Question 6

What is the proof of the properties of the inner product

Answer 6

Question 7

What is ⟨Au, v⟩ equal to

Answer 7

Question 8

Describe the proof that

Answer 8

Question 9

When are two vectors orthogonal

Answer 9

⟨u, v⟩ = 0

Question 10

What is the norm of a vector

Answer 10

Question 11

When is a set orthogonal

Answer 11

A set S ⊆ V is orthogonal if, for all u, v ∈ S
u ≠ v ⇒ ⟨u, v⟩ = 0

Question 12

Give examples of some orthogonal sets

Answer 12

Question 13

When is a set orthonormal

Answer 13

If it is orthogonal and consists only of unit vectors

Question 14

What is an orthonormal basis

Answer 14

A basis of V which is an orthonormal set

Question 15

If V is an inner product space and B = {u1, u2, . . . , un} is an
orthonormal basis for V , what can we say about every v ∈ V ,

Answer 15

Question 16

What can we say about any orthogonal set of non-zero vectors

Answer 16

It is linearly independent

Question 17

If V is a finite-dimensional inner product space and |S| = dim V what can we say about S

Answer 17

S is a basis of V

Question 18

What is the equation for the Gram-Schmidt process

Answer 18

Question 19

When is a linear transformation an isometry

Answer 19

Question 20

What is a matrix unitary

Answer 20

A matrix is said to be unitary if the columns of it form an orthonormal basis of F^n (When F = R we say that U is orthogonal).

Question 21

What is the equivalent to a matrix being unitary

Answer 21