Inner Products

Question 1

Axioms: Inner Product

Answer 1

I <v|w> = (<w|v>)*
II Linearity in the second slot + Conjugate linearity in the first slot
III <v|v> is real and >= 0
IV <v|v> = 0 then |v> = |0>

Question 2

Definition: Complex Inner Product Space

Answer 2

|v> ** |w>
or
<v|w>

Question 3

Definition: Standard inner product

Answer 3

<v|w> := |v>†|w> ie matrix multiply with first conjugated

Question 4

Definition: Norm of a vector

Answer 4

|||v>|| := Sqrt(<v|v>)

Question 5

Theorem: Inner product with zero

Answer 5

<v|0> = <0|v> = 0

Question 6

Theorem: Cauchy-Schwartz inequality

Answer 6

|<v|w>|^2 <= <v|v><w|w>

Question 7

Theorem: Inner product property

Answer 7

(A|v>) ** |w> = |v> ** (A†|w>) (El swappje)

Question 8

Definition: Orthogonal Vectors

Answer 8

<v|w> = 0

Question 9

Theorem: Scalar mult. of orthogonal vectors

Answer 9

Act like orthogonal vectors dot

Question 10

Definition: Orthonormal

Answer 10

Orthogonal plus every vector is a unit vector

Question 11

Definition: Orthonormal Basis

Answer 11

Orthonormal and is the number of dimensions with each vector

Question 12

Theorem: Determing Expansion coef.

Answer 12

|v> =
Sum: λi|vi>
then
λi = |vi> ** |v>