The problems states "All polynomials of the form p(t)= at^2, where a is in R." Why is span written as such span{t^2}. I dont understand how t^2 can be considered part of the span?
imagine that we turn the real polynomials into (an infinite-dimensional) vector space (over R) in the following way:
we use the basis: {1,t,t^{2},.......} (this is an infinite set).
so the polynomial a_{0} + a_{1}t + a_{2}t^{2}+...+ a_{n}t^{n}
has coordinates in this basis:
(a_{0},a_{1},a_{2},....,a_{n},0,0,0.....)
(everything is 0 after the n-th coordinate).
so what is the subspace span({t^{2}})?
well t^{2} is the vector: (0,0,1,0,0,0........)
so span({t^{2}}) consists of those vectors that have 0's everywhere but in the 3rd coordinate (everything but the t^{2} term is 0, the t^{2} term might be zero or non-zero).
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to avoid using "infinite dimensions", often one just considers all polynomials of degree n or less. this gives a vector space of dimension n+1. n = 2,3 and 4 are popular choices. for example, for polynomials of degree 3 or less, we can write:
t^{2} = (0,0,1,0), which stands for: t^{2} = (0)1 + (0)t + (1)t^{2} + (0)t^{3}, which is pretty obvious.