Linear Algebra 19d: Illustration of Component Spaces by a Sum of Polynomials

[tangolasher]

+MathTheBeautiful

Something I've been wondering since the start of this series - I get that for Rn, a vector (1, 2, 3) has components in three different dimensions and therefore these can't be added together as a single scalar, but in the case of polynomials, you seem to treat variables raised to a different power in the exact same way; 1, x, and x^2 live in 3 different dimensions just like (1, 2, 3).

This doesn't sit right with me, because x^2 uses the same variable x, and can be calculated if you know x. A value living in dimension 2 of Rn, on the other hand, *cannot* be calculated by knowing a value in dimension 1 - they have *totally different meanings.*

Because you can't add x and x^2 together as you would if they had the same power (x + x = 2x), I'm assuming this is your basis for calling them "linear independent." But it still seems weird to me that raising them to a different power magically puts them in a different dimension. If you graphed y = x and y = x^2, the two different graphs would look different, but still be in the same dimension on the graph paper, right? One wouldn't be 3D or sticking out of the paper, right? So how do you draw this equivalency, that the 3 dimensional vector (1, 2, 3) corresponds to (1, x, x^2)?