Let V be the vector space of all real polynomials over R. Let W be the vector space of polynomials which are divisible by x^6. Show that the quotient space V/W has dimension 6.

Question

dan815 · Answer

ah this question is quite easy

terenzreignz · Answer

What's a quotient space?  Like a quotient group?

anonymous · Answer

Pretty much the same thing.

terenzreignz · Answer

Like cosets and stuff? :D

anonymous · Answer

Yep, in this case W is called the coset of V, just like in groups.

terenzreignz · Answer

I suppose we could dream up a basis out of thin air...

terenzreignz · Answer

This basis...$$1+W\x+W\x^2+W\x^3+W\x^4+W\x^5+W$$

terenzreignz · Answer

I can only sketch it in my mind, but normally, the basis of polynomials is infinite, right?
But... if the exponent of x gets any bigger than 5, then it just reverts to one of the lesser exponents... hang on, let me say that in a manner that makes sense...

terenzreignz · Answer

For instance, x^6
$$\large x^6 + W = W$$

anonymous · Answer

I don't follow your reasoning.

anonymous · Answer

No worries, I understand what you mean now. THANKS!