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# Definition: Inner Product

You may have run across inner products, also called dot products, on $Rn n$ before in some of your math or science courses. If not, we define the inner product as follows, given we have some $x∈Rn x n$ and $y∈Rn y n$

Definition 1: standard inner product
The standard inner product is defined mathematically as:
$⟨x,y⟩=yT⁢x=( y 0 y 1 … y n − 1 )⁢ x 0 x 1 ⋮ x n − 1 =∑ i =0n−1 x i ⁢ y i x y y x y 0 y 1 … y n − 1 x 0 x 1 ⋮ x n − 1 i n 1 0 x i y i$
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# Inner Product in 2-D

If we have $x∈R2 x 2$ and $y∈R2 y 2$, then we can write the inner product as

$⟨x,y⟩=∥x∥⁢∥y∥⁢cosθ x y x y θ$
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where $θθ$ is the angle between $xx$ and $yy$.

Geometrically, the inner product tells us about the strength of $xx$ in the direction of $yy$.

Example 1

For example, if $∥x∥=1 x 1$, then $⟨x,y⟩=∥y∥⁢cosθ x y y θ$

The following characteristics are revealed by the inner product:

• $⟨x,y⟩ x y$ measures the length of the projection of $yy$ onto $xx$.
• $⟨x,y⟩ x y$ is maximum (for given $∥x∥ x$, $∥y∥ y$) when $xx$ and $yy$ are in the same direction ( $(θ=0)⇒(cosθ=1) θ 0 θ 1$).
• $⟨x,y⟩ x y$ is zero when $(cosθ=0)⇒(θ=90°) θ 0 θ 90°$, i.e. $xx$ and $yy$ are orthogonal.

# Inner Product Rules

In general, an inner product on a complex vector space is just a function (taking two vectors and returning a complex number) that satisfies certain rules:

• Conjugate Symmetry: $⟨x,y⟩=⟨x,y⟩¯ x y x y$
• Scaling: $⟨α⁢x,y⟩=α⁢⟨(x,y)⟩ α x y α x y$
• Additivity: $⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩ x y z x z y z$
• "Positivity": $∀ x ,x≠0:⟨x,x⟩>0 x x 0 x x 0$
Definition 2: orthogonal
We say that $xx$ and $yy$ are orthogonal if: $⟨x,y⟩=0 x y 0$