Dot products and duality | Essence of linear algebra, chapter 7
3.1.15-Linear Algebra: Self-Adjoint and Symmetric Matrices
Mod-01 Lec-36 L36-Spectral Theorem: http://youtu.be/vMzykSnxEUY
(U1): ∀λ∈F:∀x,y∈G: λ∘(x+Gy)=λ∘x+Gλ∘y
prove the associativity of vector addition (Property AA). This is a bit tedious, though necessary. Throughout, the plus sign (“+”) does triple-duty. You might ask yourself what each plus sign represents as you work through this proof
Theorem AIU (Additive Inverses are Unique) Suppose that V is a vector space. For each u∈V, the additive inverse, −u, is unique.
As obvious as the next three theorems appear, nowhere have we guaranteed that the zero scalar, scalar multiplication and the zero vector all interact this way. Until we have proved it, anyway.
Theorem ZSSM (Zero Scalar in Scalar Multiplication) Suppose that V is a vector space and u∈V. Then 0u=0.
Here’s another theorem that looks like it should be obvious, but is still in need of a proof.
via Vector Spaces.