Linear algebra - vector spaces, bases

[tawi]

Homework Statement

1) In a vector space V of all real polynomials of third degree or less find basis B such that for arbitrary polynomial p \in V the following applies:
[p]_B = \begin{pmatrix} p'(0)\\p'(1)\\p(0)\\p(1)\end{pmatrix} where p' is the derivative of the polynomial p.

Homework Equations

The Attempt at a Solution

Would somebody be so kind and pointed me in a direction how to solve this? I have missed couple last lectures and I can´t seem to find this specifit type of problem anywhere.. Thanks:)

 

[RUber]

Normally, the space of third degree or less polynomials would be spanned by the standard basis:
$\begin{pmatrix} 1\\0\\0\\0 \end{pmatrix},\begin{pmatrix} 0\\1\\0\\0 \end{pmatrix},\begin{pmatrix} 0\\0\\1\\0 \end{pmatrix},\begin{pmatrix} 0\\0\\0\\1 \end{pmatrix}.$
And the polynomial
$ax^3 + bx^2+ cx + d$ would be represented by $\begin{pmatrix} d\\c\\b\\a \end{pmatrix}.$
This problem is asking you for a different way to define a basis set such that the coefficient of p in first basis will always be p'(0), the coefficient of p in the second basis will always be p'(1), and so on.
Start with what you know those values have to be for a general polynomial. Then ask yourself what combinations would give you that result...finally, test to see if you indeed have a basis.

 

[tawi]

so for $p'(0)$ we'll have $c$, for $p'(1)$ we will have $3a^2+2b+c$, for $p(0) d$ and for $p(1) a^3+b^2+c+d$... How will the final solution look like then? Just a vector with these elements?

 

[epenguin]

Homework Statement

1) In a vector space V of all real polynomials of third degree or less find basis B such that for arbitrary polynomial p \in V the following applies:
[p]_B = \begin{pmatrix} p'(0)\\p'(1)\\p(0)\\p(0)\end{pmatrix} where p' is the derivative of the polynomial p.

Homework Equations

The Attempt at a Solution

Would somebody be so kind and pointed me in a direction how to solve this? I have missed couple last lectures and I can´t seem to find this specifit type of problem anywhere.. Thanks:)

Have you made an error in transcription of the problem? Maybe there should be a p(1) or something?

 

[tawi]

Yes, the last one should in fact be p(1).

 

[epenguin]

Would the question in ordinary algebra be: express any polynomial ax³ + bx² + cx + d in terms of those other things, two of which are already c and d.
That is not very difficult in ordinary algebra, so maybe you could express your result back into vector space terms, and maybe the ordinary calculation back into vector space terms.
Not being very handy with vector spaces, I would like to see it.

 

[tawi]

You mean writing $ax^3 + bx^2 + cx + d = p$ which will give us system of four equations with the same LHS and RHS will always be the result for the individual elements of the basis?
So the first equation would be $ax^3 + bx^2 + cx + d = c$, second $ax^3 + bx^2 + cx + d = d$, third $ax^3 + bx^2 + cx + d = 3a^2 + 2b + c$ and $ax^3 + bx^2 + cx + d = a^3 + b^2 + c+ d$?

 

[epenguin]

No I don't think I meant that, that doesn't seem to me to mean anything.
I did mean just obtain a, b, c, d in terms of the elements you were given.

I don't know what your conventions are for writing polynomials - RUber seems to be leaving out x's altogether.
My concrete way would be a horizontal vector
(x⁰, x¹, x²,x³) that multiplies a matrix made of RUber's columns that multiplies his vertical vector.
I could change that into a different matrix that multiplies the vector in the problem.

 

[RUber]

so for $p'(0)$ we'll have $c$, for $p'(1)$ we will have $3a^2+2b+c$, for $p(0) d$ and for $p(1) a^3+b^2+c+d$... How will the final solution look like then? Just a vector with these elements?

Note that p'(1) will not have coefficients raised to powers. The x terms were the variables raised to powers and you are plugging in x=1 to the polynomial.

Example p(1)= a(1^3)+b(1^2)+c(1)+d= a+b+c+d.