Mapping over Magnitude – Projections

Given a line ‘a’ and a point ‘b’ we want to find a scaled version(λ)’a’that is as close to ‘b’ as possible.

The right answer is where the line of λa to b is at a right angle to line a.

This line is the difference of the line λa and the line that goes to point ‘b’ , line b –>(b-λa).

proj(a)b = λa –> projecting point b onto a subspace of line a equals scaled line a.

The dot product between both vector lines is zero is the mapping over magnitude as followed:

–> Transpose(a)(b-λa) = 0

Transpose(a)b – Transpose(a)(λa)=0

Transpose(a)(λa)=Transpose(a)b .

λ = Transpose(a)b /Transpose(a)(a)

This is also called mapping over magnitude, because  Transpose(a)b  is a mapping between two vectors (also known as the dot product of two vectors)  and Transpose(a)(a) is in fact the compression scaler of the squared length/ magnitude of the line/vector a.

 

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