Does this set of polynomials span P3?

In summary: So in summary, for polynomials, we can use the same method of setting up a matrix and reducing the system to determine if a set of vectors spans the space. This can also be done by solving a system of equations using the definition of spanning.
  • #1
mitch_1211
99
1
hey i want to find out if the set
s = {t2-2t , t3+8 , t3-t2 , t2-4} spans P3

For vectors, i would setup a matrix (v1 v2 v3 v4 .. vn | x) where x is a column vector (x , y ,z .. etc) and reduce the system. If a solution exists then the vectors span the space, if there are no solutions then the space spanned is either the line or plane made up of the x , y ,z = 0

My question is, how do u setup a similar system for polynomials?

i know that the colums would be
t3
t2
t
1

and then put in the polynomials like (p1 p2 p3 p4 .. ) but what do i augment this matrix with?

many thanks

mitch
 
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  • #2
Hi,

It is an identical situation. Rather than having a vector [itex](w,x,y,z)=w \mathbf{i} + x\mathbf{j}+y\mathbf{k}+z\mathbf{l}[/itex], you have a vector [itex](w,x,y,z)=w 1 +x t + y t^2 +z t^3[/itex]. The [itex]\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l}[/itex] and the [itex]1,t,t^2,t^3[/itex] are standard basis vectors, and the [itex]w,x,y,z[/itex] are components. In this notation, your basis is [itex]\{(0,-2,1,0),(8,0,0,1),(0,0,-1,1),(-4,0,1,0)\}[/itex], a trivial change which hopefully brings you into familiar territory!
 
  • #3
So after I've got the basis vectors you have described, I augment them in a matrix with w,x,y,z just like for usual vectors?
 
  • #4
mitch_1211 said:
So after I've got the basis vectors you have described, I augment them in a matrix with w,x,y,z just like for usual vectors?

Yes!
 
  • #5
Another way to do this, without directly using matrices, is to use the definition:
The set [itex]\{t^2-2t , t^3+8 , t^3-t^2 , t^2-4\}[/itex] spans P3 if and only if, for any a, b, c, d there exist [itex]\alpha[/itex], [itex]\beta[/itex], [itex]\gamma[/itex], [itex]\delta[/itex] such that
[tex]\alpha(t^2- 2t)+ \beta(t^3+ 8)+ \gamma(t^3- t^2)+ \delta(t^2- 4)= at^3+ bt^2+ ct+ d[/tex]
which is the same as
[tex](\beta+ \gamma)t^3+ (\alpha- \gamma+ \delta)t^2- (2\alpha+ \gamma- \delta)t+ (8\beta- 4\delta)= at^3+ bt^2+ ct+ d[/tex]
That gives the four equations
[tex]\beta+ \gamma= a[/tex]
[tex]\alpha- \gamma+ \delta= b[/tex]
[tex]2\alpha+ \gamma- \delta= c[/tex]
[tex]8\beta- 4\delta= d[/tex]

The set spans the space if and only if it is possible to solve for [itex]\alpha[/itex], [itex]\beta[/itex], [itex]\gamma[/itex], and [itex]\delta[/itex] in terms of any numbers, a, b, c, and d.

Of course, solving that system of equations could be done in terms of the matrix of coefficients which gets right back to your method!
[tex]\beta+ \gam,
 
  • #6
micromass said:
Yes!

Perfect, thank you!
 
  • #7
HallsofIvy said:
Of course, solving that system of equations could be done in terms of the matrix of coefficients which gets right back to your method!

Thanks for explaining that, much appreciated.
 

Related to Does this set of polynomials span P3?

1. What does it mean for a set of polynomials to span P3?

A set of polynomials spans P3 if every polynomial in P3 can be written as a linear combination of the polynomials in the set. In other words, the set contains enough elements to represent any polynomial in P3.

2. How can I determine if a set of polynomials spans P3?

To determine if a set of polynomials spans P3, you can use the following steps:
1. Choose a random polynomial in P3.
2. Write the polynomial as a linear combination of the polynomials in the set.
3. If it is possible to write the polynomial as a linear combination of the set, then the set spans P3.

3. Can a set of polynomials span P3 if it contains less than 3 polynomials?

No, a set of polynomials must contain at least 3 polynomials to span P3. This is because P3 is the set of all polynomials of degree 3 or less, and a set with less than 3 polynomials cannot represent all possible polynomials in P3.

4. Is it possible for a set of polynomials to span P3 if the polynomials have different degrees?

Yes, it is possible for a set of polynomials to span P3 even if the polynomials have different degrees. This is because a polynomial with a lower degree can still be written as a linear combination of polynomials with higher degrees.

5. Can a set of polynomials span P3 if one of the polynomials is a multiple of another polynomial in the set?

Yes, a set of polynomials can still span P3 if one polynomial is a multiple of another polynomial in the set. This is because the multiple can be factored out of the linear combination and the remaining polynomials can still represent all polynomials in P3.

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