Is the set S a subspace of P_3

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Bronn

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I'm not even sure I'm interpreting this question properly.

Q: is the set S a subspace of P3

S = { p element of P3: p'(x) + x + 1 = 0 for all x is an element of R}



now, first of all, I'm a bit confused with the vector space of P3 . Is it the space of polynomials of degree 3 or less or do the polynomials have to have a degree 3?

first thing I tried was to let p(x) = a0 + a1x + a2x2 + a3x3 where a0,a1,a2,a3 are elements of R

then p'(x) = a1 + 2a2x + 3a3x2

so

p'(x) + x + 1 = 0

is

(a1+1) + (2a2+1)x + 3a3x2 = 0

so when x=0 (a1+1) = 0

now the answer in the back of the book is that it is not a vector space because it doesn't have a zero vector. But what if a1= -1?
I suspect I'm missing something important here

thanks
 
Bronn said:
I'm not even sure I'm interpreting this question properly.

Q: is the set S a subspace of P3

S = { p element of P3: p'(x) + x + 1 = 0 for all x is an element of R}



now, first of all, I'm a bit confused with the vector space of P3 . Is it the space of polynomials of degree 3 or less or do the polynomials have to have a degree 3?

first thing I tried was to let p(x) = a0 + a1x + a2x2 + a3x3 where a0,a1,a2,a3 are elements of R

then p'(x) = a1 + 2a2x + 3a3x2

so

p'(x) + x + 1 = 0

is

(a1+1) + (2a2+1)x + 3a3x2 = 0

so when x=0 (a1+1) = 0

now the answer in the back of the book is that it is not a vector space because it doesn't have a zero vector. But what if a1= -1?
I suspect I'm missing something important here

thanks
Click to expand...
For a quadratic equation to equal 0 we did NOT just need the constant to be 0. After all, 3x2 + 7x + 0 is NOT 0. Rather 0x2 + 0x + 0 = 0!!!

So why does S not contain the 0 vector?
Based on your answer to my question can you now decide if P3 is the space of polynomials of degree 3 or less or do the polynomials have to have a degree 3?
 
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So is it that the zero polynomial is when a0, a1,a2,a3 = 0 ?


so (0+1)+ (0+1)x + 0x 2 =/= 0 + 0 x+ 0x2

therefore S doesn't contain a zero polynomial
 
Bronn said:
So is it that the zero polynomial is when a0, a1,a2,a3 = 0 ?


so (0+1)+ (0+1)x + 0x 2 =/= 0 + 0 x+ 0x2
Click to expand...
Obviously not.You above let a0, a1,a2,a3 = 0 and concluded that it was not the zero polynomial. You need to think a bit clearer about what is going on. Please give it another try.
 
sorry, I got no idea. There's only one example in my book dealing with polynomial subspaces and for proving the zero polynomial it just states 'obviously it has one' and moves on, so I haven't had much to work with on this.
 
Bronn said:
sorry, I got no idea. There's only one example in my book dealing with polynomial subspaces and for proving the zero polynomial it just states 'obviously it has one' and moves on, so I haven't had much to work with on this.
Click to expand...
Exactly when will (a1+1) + (2a2+1)x + 3a3x2 = 0? Please reply with your answer! And polynomials in P3 must be of degree 3. This last statement is your BIG hint (so look at it for a bit)
 
maybe I see it now...


(-1+1) + (2(-.5)+1)x + 3(0)x2 = 0+0x +0x​2


so the zero polynomial is when a1 =-1 , a2 = -.5 and a3 = 0

However, when a3 = 0
p(x) is no longer a polynomial of degree 3? so it's not an element of S.
 
Bronn said:
I'm not even sure I'm interpreting this question properly.

Q: is the set S a subspace of P3

S = { p element of P3: p'(x) + x + 1 = 0 for all x is an element of R}

now, first of all, I'm a bit confused with the vector space of P3 . Is it the space of polynomials of degree 3 or less or do the polynomials have to have a degree 3?
Click to expand...

What DEFINITION were you given for P3? That is how you answer this question. Here is one page defining it, as an example; you'll have to check whether it agrees with what you were given.
 
Dr.Peterson said:
What DEFINITION were you given for P3? That is how you answer this question. Here is one page defining it, as an example; you'll have to check whether it agrees with what you were given.
Click to expand...
Dr P, you are absolutely correct. I was not thinking and it was right in front of me. How can the 0 polynomial be in P_3, if the degree of all polys in P_3 must be of degree 3? I really blew this one! So yes, the P_3 is the set of all polyn of degree <= 3.
 
Bronn said:
I'm not even sure I'm interpreting this question properly.

Q: is the set S a subspace of P3

S = { p element of P3: p'(x) + x + 1 = 0 for all x is an element of R}



now, first of all, I'm a bit confused with the vector space of P3 . Is it the space of polynomials of degree 3 or less or do the polynomials have to have a degree 3?

first thing I tried was to let p(x) = a0 + a1x + a2x2 + a3x3 where a0,a1,a2,a3 are elements of R

then p'(x) = a1 + 2a2x + 3a3x2

so

p'(x) + x + 1 = 0

is

(a1+1) + (2a2+1)x + 3a3x2 = 0

so when x=0, (a1+1) = 0

now the answer in the back of the book is that it is not a vector space because it doesn't have a zero vector. But what if a1= -1?
I suspect I'm missing something important here

thanks
Click to expand...
Let's try this again.
Actually this is trivial. The 0 polynomial in P3 = 0. The question is whether or not this zero polynomial is in S.
Well the zero polynomial, P(x), is in S iff P'(x) + x + 1 = 0
Well P'(x) =0 and P'(x) + x + 1 = 0 + x + 1 which is NOT 0, so the zero polynomial is NOT in S.

This was one of the easier proof I've done.

Note: I am giving the student the complete proof because I guided the student very poorly in my previous posts.

I'll be in the corner thinking about how sloppy I was. Yuck!
 
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Dr.Peterson said:
What DEFINITION were you given for P3? That is how you answer this question. Here is one page defining it, as an example; you'll have to check whether it agrees with what you were given.
Click to expand...

the problem is I wasn't given an explicit definition (maybe its assumed in the course) and I've been trying to piece together the definition based on examples. Which gets confusing.

I will mull over this info and try to understand it. thanks
 
Jomo said:
Let's try this again.
Actually this is trivial. The 0 polynomial in P3 = 0. The question is whether or not this zero polynomial is in S.
Well the zero polynomial, P(x), is in S iff P'(x) + x + 1 = 0
Well P'(x) =0 and P'(x) + x + 1 = 0 + x + 1 which is NOT 0, so the zero polynomial is NOT in S.

This was one of the easier proof I've done.

Note: I am giving the student the complete proof because I guided the student very poorly in my previous posts.

I'll be in the corner thinking about how sloppy I was. Yuck!
Click to expand...


this makes sense.


I had problem

show the set S={p element of P3 : p''(x)=0, for all x is an element of R}

now am I interpreting this correctly p''(x)=0 means it must be p''(x)=0+0x=0
so this can only be true for the set of polynomials degree 1 or less?

so the set S is the set of polynomials of degree 1 or less in P3 ?
 
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Bronn said:
the problem is I wasn't given an explicit definition (maybe its assumed in the course) and I've been trying to piece together the definition based on examples. Which gets confusing.

I will mull over this info and try to understand it. thanks
Click to expand...
P3 IS a vector space which means it has to have the zero polynomial and be closed under addition. Therefore P3 MUST contain polynomial of degree less than 3! I do not know what I was thinking. I am just glad you read my updated post.
 
Bronn said:
the problem is I wasn't given an explicit definition (maybe its assumed in the course) and I've been trying to piece together the definition based on examples. Which gets confusing.

I will mull over this info and try to understand it. thanks
Click to expand...

Surely something was said about what they mean by P3, or they couldn't have asked the question at all! I was asking for that. Possibly it implies the full definition (for example, it might have called P3 "the vector space of polynomials of degree 3"); or possibly the author is just very lax. Did you check the index of the book?

Also, what course is this? What prerequisite course is there?
 
this is the class

https://www.maths.unsw.edu.au/courses/math1231-mathematics-1b



No, I rechecked there is no "this is what P3 is..etc" it just shows up in an example and says "let P3 be the set of polynomials of degree 3 or less...etc" So I was confused if that's what P3 always meant or is it just for that example.
Its possible later chapters deal with them properly and this chapter touches them without much explanation.
 
Bronn said:
this is the class

https://www.maths.unsw.edu.au/courses/math1231-mathematics-1b
Click to expand...

The important thing here is that it is, in part, a course in linear algebra. I asked mostly because of your suggestion that the information might be in a prerequisite course, which turns out not to matter:

Bronn said:
No, I rechecked there is no "this is what P3 is..etc" it just shows up in an example and says "let P3 be the set of polynomials of degree 3 or less...etc" So I was confused if that's what P3 always meant or is it just for that example.
Its possible later chapters deal with them properly and this chapter touches them without much explanation.
Click to expand...

That sounds like a definition to me! That's what I was asking for. "Let __ be ___" means, "I will define __ as __". And they clearly said exactly what it is, which is what you were asking about: "degree 3 or less". You didn't need to look any further; that statement makes it clear that they haven't previously defined it.

It doesn't really matter whether this is universal or not; you may have noticed that I didn't find a standard source containing a definition, which suggests that, although it may be common as an example, it is not necessarily important enough in itself to say a lot about. All you need is a definition that tells you what they mean by it in this problem. That's how definitions work.
 
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Dr.Peterson said:
The important thing here is that it is, in part, a course in linear algebra. I asked mostly because of your suggestion that the information might be in a prerequisite course, which turns out not to matter:



That sounds like a definition to me! That's what I was asking for. "Let __ be ___" means, "I will define __ as __". And they clearly said exactly what it is, which is what you were asking about: "degree 3 or less". You didn't need to look any further; that statement makes it clear that they haven't previously defined it.

It doesn't really matter whether this is universal or not; you may have noticed that I didn't find a standard source containing a definition, which suggests that, although it may be common as an example, it is not necessarily important enough in itself to say a lot about. All you need is a definition that tells you what they mean by it in this problem. That's how definitions work.
Click to expand...


yeah, I was getting confused because some examples dealt with polynomials of exactly 3 degrees and some 3 and lower. And some examples just state such and such is an element of P3 without a definition of what they meant by P 3. So I was trying to figure out what P3 meant when its not explicitly defined in an example. A definition in the chapter content would have sorted all this out.
anyway I got it now thanks!
 
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