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Linear Transformation
A linear transformation from V to W is a function T:V→W with the following properties for any u,v∈V and c∈F
  • T(u+v)=T(u)+T(v)
  • T(cu)=cT(u)
Inner Product
An inner product is a function ⟨⋅,⋅⟩:V×V→F where V is a vector space over the field F, it assigns to each pair 𝐯,𝐰∈V a real number ⟨𝐯,𝐰⟩ such that, for all 𝐮,𝐯,𝐰∈V and α∈𝐅 and satisfies:
  1. ⟨𝐯,𝐯⟩≥0, with equality if and only if 𝐯=𝟎.
  2. ⟨𝐯,𝐰⟩=⟨𝐰,𝐯⟩.
  3. ⟨𝐮+𝐯,𝐰⟩=⟨𝐮,𝐰⟩+⟨𝐯,𝐰⟩,
  4. ⟨α𝐯,𝐰⟩=α⟨𝐯,𝐰⟩.
Inner Product Space
An inner product space is a vector space V over the field F together with an inner product
The Dot Product in Rn Forms an Inner Product
The dot product in ℝn forms an inner product
We verify the axioms in the definition of an inner product. Positive definiteness is exactly the fact that the dot product is positive definite. Symmetry is exactly the fact that the dot product is symmetric. Additivity in the first coordinate is exactly dot product distributes into vector addition from the right. Compatibility with scalar multiplication is exactly dot product compatibility with scalar multiplication. Therefore the dot product satisfies all the inner product axioms.
Schwarz Inequality
For all x,y∈ℝn we have |⟨x,y⟩|≤‖x‖‖y‖ and |⟨x,y⟩|=‖x‖‖y‖ if and only if x,y are collinear
The Norm Satisfies the Triangle Inequality
For any x,y∈ℝn we have ‖x+y‖≤‖x‖+‖y‖