lacunary - Mathnotes

Projection

Definition of Projection

Definition: projection (also: vector projection, orthogonal projection) (also defines: scalar projection, rejection) \@{projection}

Take two vectors in Rn, u and v, with v≠0.

The scalar projection (also: scalar component) of u onto b is a scalar equal to a1=‖a‖cos⁡θ, where θ is the @angle between a and b. It is denoted as follows, with an equivalent formula:

compvu=‖projvu‖=⟨u,v⟩‖v‖

The projection of u onto v is the vector whose magnitude is the scalar projection of u onto v with the same direction as v:

projvu=⟨u,v⟩‖v‖2v=(‖a‖cos⁡θ)b^.

The rejection (also: vector rejection) of u from v is the component of u that is orthogonal to v and is given as

oprojvu=u−projvu.

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Intuition \@{intuition-2}

One way to visualize this is to shine a flashlight from a direction perpendicular to v, with u between the flashlight and v (pretend the vectors are opaque). The shadow the flashlight casts onto v will be the projection of u onto v.

Theorem: Orthogonal Decomposition \@{orthogonal-decomposition}

For a,b∈Rn, with b≠0,

a=projba+oprojba.

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Theorem: Pythagorean Theorem \@{pythagorean-theorem}

For orthogonal vectors a,b∈Rn,

‖a‖2+‖b‖2=‖a+b‖2.

Proof \@{proof-of-pythagorean-theorem}

By definition of the @euclidean norm, ‖a+b‖2=⟨a+b,a+b⟩=‖a‖2+2⟨a,b⟩+‖b‖2.Because a and b are orthogonal, their @euclidean inner product is 0, and we have‖a+b‖2=‖a‖2+‖b‖2.

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Theorem: Cauchy-Schwarz Inequality \@{cauchy-schwarz-inequality}

For any vectors x,y∈Rn,

|⟨x,y⟩|≤‖x‖‖y‖,

where ⟨x,y⟩ denotes the @euclidean inner product and ‖x‖=⟨x,x⟩ is the @euclidean norm.

Proof \@{proof-of-cauchy-schwarz-inequality}

From Orthogonal Decomposition we have a=projba+oprojba.Taking the @euclidean norm squared of both sides gives ‖a‖2=‖projba+oprojba‖2=‖projba‖2+2⟨projba,oprojba⟩+‖oprojba‖2=‖projba‖2+‖oprojba‖2,with the middle term dropping out due to orthogonality. Now, because ‖oprojba‖2≥0,‖a‖2≥‖projba‖2.Now we'll flip sides and expand projection according to its definition‖projba‖2≤‖a‖2‖⟨a,b⟩‖b‖2b‖2≤(⟨a,b⟩‖b‖2)2‖b‖2≤⟨a,b⟩2‖b‖2≤‖a‖2⟨a,b⟩2≤‖a‖2‖b‖2.Taking the square root of both sides we get what we want to show:⟨a,b⟩2≤‖a‖2‖b‖2⟨a,b⟩≤‖a‖‖b‖.

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Intuition \@{cauchy-schwarz-inequality-intuition}

This is saying that the length of the projection of a onto b cannot exceed the length of a.

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