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

[seang]

Homework Statement

Determine whether the set of polynomials of degree 3 form a subspace of P(4)

Homework Equations

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

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.

 

[courtrigrad]

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

 

[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?

 

[courtrigrad]

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 $$V$$?

 

[radou]

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

Thanks, Sean.

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

 

[seang]

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

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.

 

[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.

 

[seang]

thanks : )