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?

Printable View

- October 13th 2012, 01:05 PMbonfire09vector subspace?
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?

- October 13th 2012, 01:26 PMjohnsomeoneRe: vector subspace?
- October 13th 2012, 01:31 PMbonfire09Re: vector subspace?
is it because that "a" is a scalar?

- October 13th 2012, 02:13 PMDevenoRe: vector subspace?
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).

********

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.