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?
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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?
BecauseQuote:
Originally Posted by bonfire09
, and
.
is it because that "a" is a scalar?
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,t2,.......} (this is an infinite set).
so the polynomial a0 + a1t + a2t2+...+ antn
has coordinates in this basis:
(a0,a1,a2,....,an,0,0,0.....)
(everything is 0 after the n-th coordinate).
so what is the subspace span({t2})?
well t2 is the vector: (0,0,1,0,0,0........)
so span({t2}) consists of those vectors that have 0's everywhere but in the 3rd coordinate (everything but the t2 term is 0, the t2 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:
t2 = (0,0,1,0), which stands for: t2 = (0)1 + (0)t + (1)t2 + (0)t3, which is pretty obvious.