Degree 3 Polynomials and Vector Spaces: Understanding Bases and Coordinates

[EvanQ]

So I have an assignment due in a few hours and I am pretty happy with it, aside from the fact that I am completely lost on the following section: -

The polynomials of degree 3, denoted P3, form a vector space.
1. Show that when added, two general polynomials of degree 3 will always produce another degree 3 polynomial.

Is this even a correct statement? To me it seems that two general polynomials of degree 3 will not always produce another degree 3 polynomial. ie. x^3 + (-x^3)?? Or is this just me not knowing what I am talking about?

2. Given p(x) E P3 and a, b E R, show that (a + b)p(x) = ap(x) + bp(x).

Here would I just state the laws of distribution and associativity regarding the Real numbers and polynomials here, and then go about the simple working to show this?

3. If P3 is a vector space, it must have a basis. Suggest a basis for this vector space.

Ok to just suggest the most simple basis in {1, x, x^2, x^3}?

4. Are there other bases for the space of degree 3 polynomials? If so, specify one.

And here is where I really start to get lost :( unless i could just say {-1, -x, -x^2, -x^3} :p


5. Generally speaking, when a basis for a vector space is known, every vector in that space can be written uniquely as a linear combination of the basis vectors. That is for v E V, we could write v = c1v1 + c2v2 + . . . + cnvn,
where the ci are scalars referred to as the coordinates of v with respect to the basis vectors v1, v2, . . . , vn. For all of the bases that you specified for P3 above, determine the coordinates of q(x) = (x − 1)3 + 3x − 1 with respect to the vectors in the basis.

To be honest I can't even make sense of that question.

This is my first time on this forum, and I am here as a last ditch result at getting a helping hand / some answers to these questions.

Forgive me if I have stepped over the line by asking for too much or anything like that.. thanks for your time.

 

[HallsofIvy]

First while this may be part of a physics assignment (really?) it is not "physics", it is mathematics so I am moving it to the mathematics thread.

So I have an assignment due in a few hours and I am pretty happy with it, aside from the fact that I am completely lost on the following section: -

The polynomials of degree 3, denoted P3, form a vector space.
1. Show that when added, two general polynomials of degree 3 will always produce another degree 3 polynomial.

Is this even a correct statement? To me it seems that two general polynomials of degree 3 will not always produce another degree 3 polynomial. ie. x^3 + (-x^3)?? Or is this just me not knowing what I am talking about?

What is your definition of "general third degree polynomial". I suspect it is any polynomial of the form ax³+ bx²+ cx+ d where a, b, c, d are ANY real numbers. In particular, any of a, b, c, d may be 0. If not the there would be no "0 vector" and this would not "form a vector space". That is, while (x³)+ (-x³)= 0 may not look like a "third degree polynomial", it is!

2. Given p(x) E P3 and a, b E R, show that (a + b)p(x) = ap(x) + bp(x).

Here would I just state the laws of distribution and associativity regarding the Real numbers and polynomials here, and then go about the simple working to show this?

Whether you need to explicitly state them or not (never a bad idea) depends upon what detail your teacher wants. Certainly you want to do something like (a+ b)(px³+ qx²+ rx+ s)= ... and multiply it out showing that you get (ap+ bp)x³+ ...= a(px³+ ...)+ b(px³+ ...).

3. If P3 is a vector space, it must have a basis. Suggest a basis for this vector space.

Ok to just suggest the most simple basis in {1, x, x^2, x^3}?

Yes, that works.

4. Are there other bases for the space of degree 3 polynomials? If so, specify one.

And here is where I really start to get lost :( unless i could just say {-1, -x, -x^2, -x^3} :p

Well, that would work! There are, in fact, an infinite number of bases. Any set of three polynomials that are "independent" will work. You might impress your teacher more if you gave a very different basis (for example, 1, x-1, (x-1)^3, (x-1)^3 would work) but then problem 5 will be harder!

5. Generally speaking, when a basis for a vector space is known, every vector in that space can be written uniquely as a linear combination of the basis vectors. That is for v E V, we could write v = c1v1 + c2v2 + . . . + cnvn,
where the ci are scalars referred to as the coordinates of v with respect to the basis vectors v1, v2, . . . , vn. For all of the bases that you specified for P3 above, determine the coordinates of q(x) = (x − 1)3 + 3x − 1 with respect to the vectors in the basis.

To be honest I can't even make sense of that question.

Did you mean (x- 1)^3+ 3x- 1? Multiply that out: x^3- 3x^2+ 3x- 1+ 3x- 1= x^3- 3x^2+ 6x- 2. Since your first basis suggested was just 1, x, x^2, x^3 it's easy to see what the coefficients must be. Your second (rather trivial) basis was -1, -x, -x^2, -x^3. What must a, b, c, d be so that x^3- 3x^2+ 6x- 2= a(-x^3)+ b(-x^3)+ c(-x)+ d? My suggestion was 1, x-1, (x-1)^2, (x-1)^3. Can you find numbers a, b, c, d so that (x- 1)^3+ 3x- 1= a(x-1)^3+ b(x-1)^3+ c(x-1)+ d? That's easier than you might think.

This is my first time on this forum, and I am here as a last ditch result at getting a helping hand / some answers to these questions.

Forgive me if I have stepped over the line by asking for too much or anything like that.. thanks for your time.

Next time, don't leave it until the last minute!

 

[EvanQ]

thanks so much for that... great great help, think i got all of it out and correct thanks to you.

 

[EvanQ]

oh and ps. it's a mathematical sciences course sorry :p just wrote physics as it's my major.