HELP - Polynomials Subspaces Question s

**angelme** · September 21st 2012, 07:34 AM

This is a proof question I have no idea what to begin with...someone please make it possible
Question 1: Are the following sets of polynomials subspaces of all polynomials of degree less than or equal to 4?

a) Even polynomials of degree less than or equal to 4
b) Odd polynomials of degree less than or equal to 4
c) Polynomials of degree 4

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Question 2: Give two polynomials that span the space of all polynomials in one variable x of degree less than or equal to 1

**FernandoRevilla** · September 21st 2012, 09:03 AM

Originally Posted by angelme

Question 1: Are the following sets of polynomials subspaces of all polynomials of degree less than or equal to 4? a) Even polynomials of degree less than or equal to 4

Denote $E=\{f\in \mathbb{R}_4[x]:f(x)=f(-x)\;\;\forall x\in \mathbb{R}\}$ the set of all even polynomials of degree less than or equal to 4. Trivially the zero polynomial belongs to $E.$ Besides, for all $\lambda,\mu\in\mathbb{R},$ for all $p,q\in E$ and for all $x\in\mathbb{R}:$

$\displaystyle\begin{aligned}(\lambda p+\mu q)(-x)&=\lambda p(-x)+\mu q(-x)\\&=\lambda p(x)+\mu q(x)\\&=(\lambda p+\mu q)(x)\\&\Rightarrow \lambda p+\mu q\in E\end{aligned}$

This means that $E$ is a subspace of $\mathbb{R}_4[x].$

b) Odd polynomials of degree less than or equal to 4

Similar arguments.

c) Polynomials of degree 4

Notice that the zero polynomial has not degree 4.

Question 2: Give two polynomials that span the space of all polynomials in one variable x of degree less than or equal to 1

Choose for example $1$ and $x.$

**angelme** · September 21st 2012, 09:35 AM

Thank you FernandoRevilla

I still don't get it. I am not good in vector space at all. Could you please give examples in numbers?

**Deveno** · September 21st 2012, 12:47 PM

unfortunately, angelme, those ARE the answers. vectors do not have to be "things with numbers". in these particular questions, the "daddy vector space" is:

$P_4 = \{ f \in \mathbb{R}[x] : \operatorname{deg}(f) \leq 4\}$

$= \{ f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 : a_0,a_1,a_2,a_3,a_4 \in \mathbb{R} \}$

now one CAN regard the polynomial $a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4$ as "being represented by the coordinates" $(a_0,a_1,a_2,a_3,a_4)$ but this requires a choice of a specific basis set, namely:

$B = \{1,x,x^2,x^3,x^4\}$ .

so, for example, the polynomial $3x^2 + 2x^4$ (which is even) has coordinates (in the basis B) (0,0,3,0,2).

the trouble is, it's not obvious from looking at the coordinates that you have an even function (that is, that f(-x) = f(x)). while this problem COULD be done that way for THIS particular basis, it's "conceptually simpler" just to go directly to the "subspace test":

a subset S of a vector space V is a subspace if and only if:

1. S is non-empty (or, equivalently: S contains the 0-vector).
2. if u and v are in S, so is their sum.
3. if v is in S, and a is a scalar, then av is in S.

conditions 2&3 may be combined into:

2A. if u,v are in S, and a,b are scalars, then au+bv is in S (which is what Fernando has done).

in general, vector spaces can be anything in which we have "things we can 'add' (the vector addition)" and things we can "stretch/shrink by a scalar" (which is why they are called "scalars"...they SCALE things), provided the two operations (vector addtion and scalar multiplication) are COMPATIBLE (the precise meaning of "compatibility" is given by the vector space AXIOMS).

the simplest EXAMPLE of a vector space is Rⁿ (often n = 2 or 3), in which vectors are "n-tuples of real numbers" so that:

x = (x₁,...,x_n), y = (y₁,...,y_n) and we have:

x+y = (x₁+y₁,...,x_n+y_n)

ax = (ax₁,...,ax_n).

but the simplicity of this example is perhaps misleading, not all vector spaces are so "nice". for example, the set of all continuous functions on a real interval [a,b] is also a vector space, and it is not so easily described by "coordinates". that's why the axioms are SO important..they aren't just "abstract rules" that one should accept blindly, they are the DEFINING properties that create the vector space structure. and when considering whether or not something IS a vector space, one often has little choice but to check the axioms.

so, you might say, we have 10 axioms, but why is it that we don't have to check all 10 (but only 3 conditions) for a subspace?

well, many of the axioms start with something like:

"for all u,v,w in V....."

well, if something is true for EVERYTHING in V, it's also true for some things that are in a SUBSET of V. so if S is a subset of V, it automatically "inherits" many of V's properties. the 3 conditions we have to check, are precisely the axioms which "aren't" inherited.

**angelme** · September 21st 2012, 06:24 PM

Thank you guys that's helpful but I still don't know what examples I should explain for question 2

Question 2: Give two polynomials that span the space of all polynomials in one variable x of degree less than or equal to 1

and how about Give 3 polynomials that span the space of all polynomials in 2 variable x and y of degree less than or equal to 1?

**FernandoRevilla** · September 21st 2012, 09:03 PM

Originally Posted by angelme

and how about Give 3 polynomials that span the space of all polynomials in 2 variable x and y of degree less than or equal to 1?

Every polynomial in $2$ variables $x$ and $y$ of degree less than or equal to 1 has the form $a+bx+cy$ so, the set $\{1,x,y\}$ spans the given subspace.

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