I'm a noob in linear algebra, following this series:
I'm extremely confused by the equivalence they draw between vectors in $\mathbb R^n$ and polynomials. Can someone please point out the flaws in my understanding (explained below)?
So for $\mathbb R^n$, a vector $(1, 2, 3)$ has components in three different dimensions (along three different axes), so adding these together is impossible, right?
And when we express a system of linear equations in matrix form, say
$2x + 3y + 10z = 45$
we treat the variables $x, y, z$ as $3$ different dimensions that cannot be directly added together, just like the different dimensions ("axes") of the vector $(1, 2, 3)$ in $\mathbb R^n$, right?
My understanding (please correct me if I'm wrong) is that $x, y, z$ somehow correspond to DIFFERENT dimensions, just as $1, 2, 3$ lie on three different axes, so you can't just add them together like $(1,2,3) = 1+2+3 = 6$, nor can you do $x+y+z = 3x$, for example.
But in the series I've been watching, I see him treat the same variable $x$, but raised to a different degree, as different variables, exactly as in the previous case. So for
$2x^3 + 3x^2 + 10x = 45$
he would make a matrix with three columns, corresponding to the $x^3, x^2,$ and $x$, exactly as if it had been
$2x + 3y + 10z = 45$
But this feels totally wrong to me, because $x^3$ and $x^2$ all come from x!
My understanding is that you can't just add variables of a different power together in a linear way, and that's why they can be slotted into the matrix in this way? i.e.)
$2x^2 + 3x^2 = 5x^2$ is okay but $2x^2 + 3x^3$ cannot be combined.
But it still feels like a stretch to me and is completely baffling conceptually. How can you say that the same variable raised to different degrees lives in different dimensions in the same way as the components of a vector in Rn??? For one, $x^2$ can be calculated if you know x, but a value living in dimension $2$ of $\mathbb R^n$, can never be calculated by just knowing a value in dimension 1 - they have totally different meanings.
If you graphed $y = x$ and $y = x^2$, the two different graphs would look kind of 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 $(x, x^2, x^3)?$