Components of a Vector  

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I took a month long break, so I thought I would make a quick forum post. Vectors can, in some sense, be described as a tuple of real numbers. Don't think this as "all vectors are tuples of real numbers," but instead "one way to look at vectors is tuples of real numbers." A vector dis still an element of a Vector Space described in my post "The Real Definition of a Vector Space," but now we will describe something called a basis. You've probably heard of these, but now we will make this more general. We say a subset B ⊂ V, for some Vector Space (V, +, ·), is a basis if ∀ v ∈ V ∃finite {ei}i=1n ⊂ B ∧ {αi}i=1n ⊂ ℝ such that α1e1 + ... + αn = v.

 

What this means is that there is a set which has a span (all linear combinations) that covers the entire Vector Space, with infinitely many zeros for each vector. From this, if B is finite, then the dimension of the Vector Space is the number of vectors in B, say n. Then, given this finite basis, we can define the components of a vector v:

 

Given the constants i}i=1n such that Σ αiei = v. Then, we can associate v with the tuple 1, α2, ... , αn). Thus, we have our definition of the components.

That one guy that's always on

 
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