# Subspaces of P spaces and C[a,b] spaces

1. Dec 5, 2006

### seang

1. The problem statement, all variables and given/known data
Determine whether the set of polynomials of degree 3 form a subspace of P(4)

2. Relevant equations

$$P(4) = c_3 x^3 + c_2 x^2 + c_1 x + c_0$$

3. The attempt at a solution

$$\alpha P(4_1) = \alpha c_3 x^3 + \alpha c_2 x^2 + \alpha c_1 x + \alpha c_0$$

This just scales the coefficients, right? It would still be a polynomial of degree 3 I think...

For addition, wouldn't you just be adding two polynomials? So wouldn't you just obtain another degree 3 polynomial?

Thanks, Sean.

Last edited: Dec 5, 2006
2. Dec 5, 2006

Note that $$\Re^{2}$$ is not a subspace of $$\Re^{3}$$

3. Dec 5, 2006

### seang

I don't see how that helps. I suspect I will, but, why is what you said true? Why don't the set of vectors (a,b,0) doesn't span a subspace in R3?

4. Dec 5, 2006

Look at vector addition and scalar multiplication in $$\Re^{2}$$ and $$\Re^{3}$$. What are the conditions to show that if $$W$$ is a nonempty subset of $$V$$ if $$V$$ is a vector space over $$\Re$$ then $$W$$ is a subspace of $$V$$?

Last edited: Dec 5, 2006
5. Dec 5, 2006

Imagine two polynomials of the third degree with leading coefficients a and -a. What happens is you add them?

6. Dec 5, 2006

### seang

Hmm. Do you mean W is nonempty, the sum of two vectors in W must lie in W, and W times a scalar must lie in W?

So take two vectors in R2, say (x1,x2),(y1,y2). Their sum equals (x1+y1,x2+y2), which lies in R2, and (cx1,cx2) also lies in R2.

7. Dec 5, 2006

### matt grime

A polynomial of degree 3 is a cubic, ax^3+bx^2+cx+d where a must be non-zero. Thus the space fails to be a vector subspace of the polys for lots of reasons. Note that the space of polynomials of degree at most 3 *is* a subspace, but this is strictly different from the space of polys of degree 3.

8. Dec 5, 2006

thanks : )