Let W be the subspace of P3(all third degree polynomials) such that p(0)=0, and let U be the subspace of all polynomials such that p(1)=0. Find a basis for W, a basis for U, and a basis for their intersection W and U.
I suppose you mean by $\displaystyle P_3$ the vector space of all polynomials of degree $\displaystyle \leq 3$. Choose a generic polynomial $\displaystyle p(x)=ax^3+bx^2+cx+d$ then, $\displaystyle p\in W\Leftrightarrow p(0)=0\Leftrightarrow d=0$ and $\displaystyle p\in U\Leftrightarrow p(1)=0\Leftrightarrow a+b+c+d=0$ . Could you continue?
To Fernando:
I'm not sure, how to continue. Should I set this arbitrary polynomial in a matrix?
If "d" is always zero for W, then the basis has to include only the variables x, x^2, x^3?
As for U, I have no clue, how to set a basis. There can be a large amount of combinations of a+b+c+d that yield 0.
here is my claim: dim(U) = 3. how can i prove this? i shall exhibit a basis. so i need to find 3 linearly independent polynomials p,q,r such that p(1) = q(1) = r(1) = 0.
for p, i choose p(x) = x-1. for q, i choose q(x) = (x-1)^2. for r, i choose (x-1)^3. are these linearly independent?
suppose ap + bq + cr = 0. that means: a(x-1) + b(x^2 - 2x + 1) + c(x^3 - 3x^2 + 3x - 1) is the 0-polynomial. now,
a(x-1) + b(x^2 - 2x + 1) + c(x^3 - 3x^2 + 3x - 1) = ax - a + bx^2 - 2bx + b + cx^3 - 3cx^2 + 3cx - c
= cx^3 + (b - 3c)x^2 + (a - 2bc + 3c)x - (a - b + c). if this is the 0-polynomial, we have to have c = 0. so
bx^2 + (a - 2b)x - (a - b) = 0. from this, we see that b = 0, as well. so:
ax - a = 0, which is only identically the 0-polynomial when a = 0.
how does this relate to my hint? if a+b+c+d = 0, if we choose any 3 numbers for a,b and c, then we are forced to set d = -a - b - c.
this suggests that picking 3 coefficients determines an element of U. the three basis vectors i chose correspond the the following choices for a,b,c,d:
x - 1 (a = 0, b = 0, c = 1, d = -1)
(x-1)^2 = x^2 - 2x + 1 (a = 0, b = 1, c = -2, d = 1).
(x-1)^3 = x^3 - 3x^2 + 3x - 1 (a = 1, b = -3, c = 3, d = -1).
i could have chosen p,q, and r quite differently. suppose i just worked from d = -a - b - c.
one "natural" choice might be: a = 1, b = c = 0, d = -1 ---> r(x) = x^3 - 1; a = c = 0, b = 1, d = -1 --> q(x) = x^2 - 1;
a = b = 0, c = 1, d = -1 --> p(x) = x - 1. you may verify for yourself that {x - 1, x^2 - 1, x^3 - 1} is also a basis for U.
now for W, it is clear that {x,x^2,x^3} is a linearly independent set, and that span({x, x^2, x^3}) is clearly a subspace of W.
suppose p(x) is in W. since p(0) = 0, p has a 0 constant term. hence p(x) = ax^3 + bx^2 + cx, which is in span({x, x^2, x^3}),
so {x, x^2, x^3} is a linearly independent spanning set for W. what do we call a linearly independent spanning set?
it would be nice if the bases we had exhibited so far had a nice intersection. but, oh well. to find a basis for U∩W, my advice is
to think about how you can factor polynomials for which p(0) = 0, and p(1) = 0. another approach:
dim(U) = dim(W) = 3. since neither U nor W is contained in the other, dim(U∩W) < 3, which leaves 0,1, or 2.
x^2 - x is in U∩W, so dim(U∩W) > 0. can you find another element of U∩W linearly independent from x^2 - x?
When we have the implicit equations of a subspace, it is just routine to find a basis of the subspace, you only need to solve the system . Working in coordinates with respect to $\displaystyle B=\{x^3,x^2,x,1\}$ we have
Basis of $\displaystyle W$
$\displaystyle W \equiv (a,b,c,d)=a(1,0,0,0)+b(0,1,0,0)+c(0,0,1,0)$
so, $\displaystyle \{(1,0,0,0),(0,1,0,0),(0,0,1,0)\}$ are linearly independent an span $\displaystyle W$ . As a consequence a basis of $\displaystyle W$ is $\displaystyle B_W=\{x^3,x^2,x\}$ .
Basis of $\displaystyle U$
$\displaystyle U \equiv (a,b,c,d)=a(1,0,0,-1)+b(0,1,0,-1)+c(0,0,1,-1)$
so, $\displaystyle \{(1,0,0,-1),(0,1,0,-1),(0,0,1,-1)\}$ are linearly independent an span $\displaystyle U$ . As a consequence a basis of $\displaystyle U$ is $\displaystyle B_U=\{x^3-1,x^2-1,x-1\}$
Basis of $\displaystyle U\cap W$
$\displaystyle U\cap W \equiv \begin{Bmatrix} a +b+c+d=0\\d=0\end{matrix}\Leftrightarrow \ldots$
$\displaystyle \Leftrightarrow (a,b,c,d)=b(-1,1,0,0)+c(-1,0,1,0)$
so, $\displaystyle \{(-1,1,0,0),(-1,0,1,0)\}$ are linearly independent an span $\displaystyle U\cap W$ . As a consequence a basis of $\displaystyle U\cap W$ is $\displaystyle B_{U\cap W}=\{-x^3+x^2,-x^2+x\}$
I understand that U can be expressed as {(1,0,0,-1),(0,1,0,-1),(0,0,1,-1)} and this vectors are linearly independent and span the polynomial.
I've been taught to put this vectors in a matrix and solve in row echelon or reduced row-echelon to find a basis, but I don't understand your methods to get the basis of U.
Could you develop further?
You have a+ b+ c+ d= 0 and can solve that for any one component- for example, d= -a- b- c. Then any vector is of the form <a, b, c, d>= <a, b, c, -a- b- c>= <a, 0, 0, -a>+ <0, b, 0, -b>+ <0, 0, c, -c>= a<1, 0, 0, -1>+ b<0, 1, 0, 1>+ c<0, 0, 1, -1>.
in P3, x^3 - 1 is the vector (1,0,0,-1) relative to the basis {x^3,x^2,x,1}: x^3 - 1 = (1)x^3 + (0)x^2 + (0)x + (-1)1.
similarly x^2 - 1 is the vector (0,1,0,-1) relative to that same basis, and x - 1 is (0,0,1,-1).
ax^3 + bx^2 + cx + d <--> (a,b,c,d).
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Jav,
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