Consider the subset U ⊂ R3[x] defined as

  • Thread starter Karl Porter
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In summary: That the sum of the two polynomials, ##(a_3x^3 + a_2x^2 + a_1x + a_0)## and ##(b_3x^3 + b_2x^2 + b_1x + b_0)##, is a degree three (or fewer) polynomial with roots at 0 and -1.
  • #1
Karl Porter
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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?
 
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  • #2
Let me try to get the notation clear first.

The sub-space you are talking about is the space of degree 3 (or lower) polynomials that have x=0 and x=-1 as roots.
The space you are talking about is the space of degree 3 (or lower) polynomials.

You are considering these as vector spaces, yes? So you want to prove closure under vector addition and upon multiplication by a scalar.

So, how would you go about proving either one of those for the sub-space?
 
  • #3
yea that's right
so for addition the sum of two vectors must belong to R3? when I substitute in -1 I get the equation a1+a3=a2+a0 but that leads me to ask how do i know a2+a0 is part of R3? is it a part of R3 because they are real numbers?
 
  • #4
Karl Porter said:
yea that's right
so for addition the sum of two vectors must belong to R3? when I substitute in -1 I get the equation a1+a3=a2+a0 but that leads me to ask how do i know a2+a0 is part of R3? is it a part of R3 because they are real numbers?
Remember what you are trying to prove.

You are not trying to prove that the sum of two polynomials with roots at 0 and -1 is a polynomial. You already know that. It follows from the fact that third degree (or fewer) polynomials are a vector space in the first place.

You are trying to prove that the sum of two polynomials, each with roots at 0 and -1 is a polynomial with roots at 0 and -1.
 
  • #5
jbriggs444 said:
Remember what you are trying to prove.

You are not trying to prove that the sum of two polynomials with roots at 0 and -1 is a polynomial.

\You are trying to prove that the sum of two polynomials with roots at 0 and -1 is a polynomial with roots at 0 and -1.
so i would add the equation to itself but one would be x1 and the other x2 and see if its gets me a polynomial with those roots.
however i don't think i can solve that because it would have two unknowns
 
  • #6
Karl Porter said:
so i would add the equation to itself but one would be x1 and the other x2 and see if its gets me a polynomial with those roots.
however i don't think i can solve that because it would have two unknowns
No. I do not think that you are getting it.

You are adding two polynomials to one another. Polynomials. Not values of the formal parameter x. You are not solving for anything. You are just checking to see if you get a result that has the right property. i.e. that it has roots at 0 and at -1.

The two polynomials would be, for instance, ##(a_3, a_2, a_1, a_0)## and ##(b_3, b_2, b_1, b_0)##.
 
  • #7
jbriggs444 said:
No. I do not think that you are getting it.

You are adding two polynomials to one another. Polynomials. Not values of the formal parameter x. You are not solving for anything. You are just checking to see if you get a result that has the right property. i.e. that it has roots at 0 and at -1.

The two polynomials would be, for instance, ##(a_3, a_2, a_1, a_0)## and ##(b_3, b_2, b_1, b_0)##.
but what happens to the x values?
 
  • #8
Karl Porter said:
but what happens to the x values?
What is the sum of ##a_3x^3 + a_2x^2 + a_1x + a_0## and ##b_3x^3 + b_2x^2 + b_1x + b_0## ?

[One step at a time. We will worry about x in a moment.]
 
  • #9
x^3(a_3+b_3)+x^2(a_2+b_2)+x(a_1+b_1)+(a_0+b_0)
 
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  • #10
Karl Porter said:
x^3(a_3+b_3)+x^2(a_2+b_2)+x(a_1+b_1)+(a_0+b_0)
Perfect. Now two questions:

1. Is this a degree three (or fewer) polynomial?
2. Does it have roots at x=0 and at x=-1?
 
  • #11
jbriggs444 said:
Perfect. Now two questions:

1. Is this a degree three polynomial?
2. Does it have roots at x=0 and at x=-1?
yes degree 3
how do i know if it has those roots? i am left over with the constant values a_0+b_0?
 
  • #12
Karl Porter said:
yes degree 3
how do i know if it has those roots? i am left over with the constant values a_0+b_0?
Suppose that the two polynomials that you just finished adding together were members of the sub-space.

That means that ##a_3x^3 + a_2x^2 + a_1x + a_0 = 0## when x=0 and when x=-1
That means that ##b_3x^3 + b_2x^2 + b_1x + b_0 = 0## when x=0 and when x=-1.

What do those facts then allow you to prove about the sum you just wrote down?
 
  • #13
a_0 =0 and b_0=0?
 
  • #14
Karl Porter said:
a_0 =0 and b_0=0?
More basic.

You have two polynomials that are both zero at x=0. You add them together. What is their sum at x=0?
You have two polynomials that are both zero at x=1. You add them together. What is their sum at x=1?
 
  • #15
sum is 0
 
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  • #16
Karl Porter said:
sum is 0
Right. So is the sum an element of the sub-space?

1. Is it a polynomial?
2. It is zero at x=0 and at x=-1?
 
  • #17
jbriggs444 said:
Right. So is the sum a member of the sub-space?

1. Is it a polynomial?
2. It is zero at x=0 and at x=-1?
ok so the sum of the two polynomials will equal 0 which is part of the subspace. and since the two polynomials add to give a polynomial to degree 3 its part of R3
 
  • #18
Karl Porter said:
ok so the sum of the two polynomials will equal 0 which is part of the subspace. and since the two polynomials add to give a polynomial to degree 3 its part of R3
Yes. Though I would state it a bit more carefully. The sum of the two polynomials is not the zero polynomial. The sum of the two polynomials is a polynomial which when evaluated at x=0 and at x=-1 yields a result of zero.
 
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  • #19
jbriggs444 said:
Yes. Though I would state it a bit more carefully. The sum of the two polynomials is not the zero polynomial. The sum of the two polynomials is a polynomial which when evaluated at x=0 and at x=-1 yields a result of zero.
would method be the same for proving its closed under multiplication? wouldn't that give a polynomial to degree 6?
 
  • #20
Karl Porter said:
would method be the same for proving its closed under multiplication? wouldn't that give a polynomial to degree 6?
Look back at post #2. We are not talking about multiplying two polynomials by each other. That is not one of the defined operations for a vector space.

The defined operation for a vector space is the multiplication of a vector by a scalar. i.e. multiplication of a polynomial by a real-valued constant.

On the other hand, perhaps you did not mean to be working with vector spaces. Perhaps you want to work in the ring of formal polynomials or in the ring of polynomial functions over the reals.
 
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  • #21
jbriggs444 said:
Look back at post #2. We are not talking about multiplying two polynomials by each other. That is not one of the defined operations for a vector space.

The defined operation for a vector space is the multiplication of a vector by a scalar. i.e. multiplication of a polynomial by a real-valued constant.
righhhttt ok so that would give a polynomial to R3 but what about the zero root?
just same as before at x=0 and x=-1?
 
  • #22
Karl Porter said:
righhhttt ok so that would give a polynomial to R3 but what about the zero root?
just same as before at x=0 and x=-1?
Assuming that we are working in the vector space of degree 3 polynomials [as opposed to the ring of formal polynomials or the ring of polynomial functions over the reals] then we can proceed as before.

Suppose that we have a real-valued constant ##b## and a polynomial ##a_3x^3 + a_2x^2 + a_1x + a_0## which is a member of the sub-space.

1. What is the product of that scalar and that polynomial?
2. Does that product evaluate to zero at x=-1 and at x=0? [editted to correct typo in original]
 
Last edited:
  • #23
K(a_3x^3+a_2x^2+a_1x+a_o)
and at x=0 x=-1 its gives 0
 
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  • #24
Karl Porter said:
K(a_3x^3+a_2x^2+a_1x+a_o)
and at x=0 x=-1 its gives 0
Good. So on that basis, is the product a member of the sub-space?
 
  • #25
jbriggs444 said:
OK. So on that basis, is the product a member of the sub-space?
yes!is that all that there is to prove to show its part of the vector space?
 
  • #26
Karl Porter said:
yes!is that all that there is to prove to show its part of the vector space?
Well, we've proved closure over addition and under multiplication by a scalar.

Let me take a quick trip to Google and see what other axioms we need...

We need the existence of a zero element in the sub-space. This still needs to be proved.

Everything else, it seems that we inherit because we are a sub-space that is closed under addition and multiplication by a scalar.
 
  • #27
jbriggs444 said:
Well, we've proved closure over addition and under multiplication by a scalar.

Let me take a quick trip to Google and see what other axioms we need...

We need the existence of a zero element in the sub-space. This still needs to be proved.

Everything else, it seems that we inherit because we are a sub-space
so multiplying the polynomial by 0 would give a zero vector right?
 
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  • #28
Karl Porter said:
so multiplying the polynomial by 0 would give a zero vector right?
Yes. But you have to make sure that the zero vector is a member of the subspace...

... oh right. Since we've already proved closure under multiplication by a scalar, that one we get for free.
 
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  • #29
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?
 
  • #30
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?
Can you justify the approach that you are taking here?

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.

You do understand the relationship between "dimension" and a "basis" for a vector space?
 
  • #31
jbriggs444 said:
Can you justify the approach that you are taking here?

Personally, I would be thinking in terms of interpolating polynomials as a way to come up with a basis for the vector space and for the sub-space.
So changing it to matrix form is to calculate the basis?
I think i went about that the wrong way, did look through some similar questions and they focus on the degree of the power of the polynomial and since every polynomial has degree ≤ 3 it has a dimension of 4?dimension is the size of the vector?
I don't really understand the basis but its spans the vector and is linearly independent
 
  • #32
Karl Porter said:
So changing it to matrix form is to calculate the basis?
That is not the way I understand things, no.

A basis is a set of vectors, each of which is a member of the space. It has two important properties:

1. The elements of the basis are linearly independent. None of them can be formed as a linear combination of the others.

2. The elements of the basis "span" the vector space. Any vector in the space can be formed as a linear combination of the basis elements.

Is this a definition that you have been exposed to?There is no guarantee of uniqueness. A vector space will not, in general, have a single unique basis. For instance, in our familiar three dimensional space, we can choose a basis of { "one meter east", "one meter north" and "one meter up" }. Or we could choose a basis of { "one foot east", "one furlong northwest", "one kilometer down" }.
 
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  • #33
by linear combination are those under +,-,x,÷ and sqrt
 
  • #34
Karl Porter said:
by linear combination are those under +,-,x,÷ and sqrt
A "linear combination" is the sum of some scalar multiple of each of the basis vectors.

Going back to our familiar three dimensional space with a basis of "one meter east", "one meter north" and "one meter up". A linear combination would be something like "2 meters east, 3 meters north, 4 meters up" with 2, 3 and 4 being the scalar multiples selected.

If your textbook does not cover this, it should. Wiki does. https://en.wikipedia.org/wiki/Vector_space#Basis_and_dimension
 
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  • #35
rightt so the basis is the 1,1,1?
 
<h2>1. What does the subset U ⊂ R3[x] represent?</h2><p>The subset U ⊂ R3[x] represents a subset of the set of polynomials with coefficients in the field of real numbers, specifically in three variables.</p><h2>2. How is the subset U ⊂ R3[x] defined?</h2><p>The subset U ⊂ R3[x] is defined as a set of polynomials in three variables with coefficients in the field of real numbers. In other words, it is a subset of the set of all polynomials in three variables with real coefficients.</p><h2>3. What is the significance of considering this subset?</h2><p>Considering this subset allows for a more specific analysis of polynomials in three variables with real coefficients. It can also be useful in solving certain types of problems or equations that involve polynomials in three variables.</p><h2>4. How is this subset different from the set of all polynomials in three variables?</h2><p>This subset is different from the set of all polynomials in three variables because it only includes polynomials with coefficients in the field of real numbers, while the set of all polynomials in three variables includes polynomials with coefficients in any field.</p><h2>5. Can you provide an example of a polynomial in U ⊂ R3[x]?</h2><p>One example of a polynomial in U ⊂ R3[x] is 2x^2 + 3xy + 5yz + 4x + 2y + 1. This polynomial has coefficients in the field of real numbers and is in three variables (x, y, and z).</p>

1. What does the subset U ⊂ R3[x] represent?

The subset U ⊂ R3[x] represents a subset of the set of polynomials with coefficients in the field of real numbers, specifically in three variables.

2. How is the subset U ⊂ R3[x] defined?

The subset U ⊂ R3[x] is defined as a set of polynomials in three variables with coefficients in the field of real numbers. In other words, it is a subset of the set of all polynomials in three variables with real coefficients.

3. What is the significance of considering this subset?

Considering this subset allows for a more specific analysis of polynomials in three variables with real coefficients. It can also be useful in solving certain types of problems or equations that involve polynomials in three variables.

4. How is this subset different from the set of all polynomials in three variables?

This subset is different from the set of all polynomials in three variables because it only includes polynomials with coefficients in the field of real numbers, while the set of all polynomials in three variables includes polynomials with coefficients in any field.

5. Can you provide an example of a polynomial in U ⊂ R3[x]?

One example of a polynomial in U ⊂ R3[x] is 2x^2 + 3xy + 5yz + 4x + 2y + 1. This polynomial has coefficients in the field of real numbers and is in three variables (x, y, and z).

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