IMO, the thread has gone astray from the original intent in post #1, which I have copied below.
Karl Porter said:
Homework Statement:: Show that U is a subspace of R3[x]
Relevant Equations:: U = {p(x) = a3x^3 + a2x^2 + a1x + a0 such that p(0) = 0 and p(−1) = 0}
so to show its a subspace (from definition) I need to prove its closed under addition and multiplication , contains 0 and for every w there is a -w? has it already been proven to contain 0 as p(-1)=0?
also I did sub in -1 and ended up with the equation a1+a3=a2+a0 but I don't know if that is relevant to solving this question?
Actually, you get two equations from the facts that p(0) = 0 and p(-1) = 0, but more about that later.
jbriggs444 said:
Perhaps you want to work in the ring of formal polynomials or in the ring of polynomial functions over the reals.
I doubt very much that rings of polynomials or polynomial functions are germane to this question, something that you (
@jbriggs444) suspected in a post that followed this quote.
Karl Porter said:
the next part asked for the dimension of U
the lack of constants is throwing me off but I put in the values of -1 and 0 and I am left with a matrix
-1 1 -1 1 │ 0
0 0 0 0 │0
so wouldn't the dimension just be 1x4?
Two points here:
1) the first row above is correct, but the second is not.
From the equation p(0) = 0, you get ##a_3 \cdot 0 + a_2 \cdot 0 + a_1 \cdot 0 + a_0 = 0##, or more simply, ##1a_0 = 0##. This means that the second row of your matrix needs to be 0 0 0 1 | 0.
2) The rank of the corrected matrix is 2. If you row-reduce that matrix, you get a basis for your function space.
jbriggs444 said:
Personally, I would be thinking in terms of interpolating polynomials as a way to come up with an alternate basis for the vector space and for the sub-space.
This might be an approach, but there is one that is much simpler, based on the matrix derived from the equations p(-1) = 0 and p(0) = 0.
The corrected matrix is
##\begin{bmatrix} -1 & 1 & -1 & 1 & | & 0 \\ 0 & 0 & 0 & 1 & | & 0\end{bmatrix}##
We can simplify this a bit by multiplying the top row by -1, to get:
##\begin{bmatrix} 1 & -1 & 1 & -1 & | & 0 \\ 0 & 0 & 0 & 1 & | & 0\end{bmatrix}##
By adding the 2nd row to the first, we get the matrix in reduced row-echelon form.
##\begin{bmatrix} 1 & -1 & 1 & 0 & | & 0 \\ 0 & 0 & 0 & 1 & | & 0\end{bmatrix}##
The first and second rows represent these equations:
##a_3 - a_2 + a_1 ...= 0##
##.....a_0 = 0##
So ##a_3## depends on ##a_2## and ##a_1##, which means that ##a_2## and ##a_1## can be chosen arbitrarily. ##a_0## depends on no other parameter, and is zero.
So
##a_3 = 1a_2 - 1a_1##
##a_2 = 1a_2 + 0a_1## (It's arbitrary.)
##a_1 = 0a_2 +1a_1## (Also arbitrary.)
##a_0 = 0a_2 + 0a_1##
If you squint your eyes a bit, you might be able to spot the coordinates of two functions that form a basis for U.
Karl Porter said:
but yes we can move onto interpolating.
I don't think this will be helpful in what you're trying to do