# More on vector spaces

1. Feb 21, 2006

### stunner5000pt

Determine if this is a vector space with the indicated operations

the set of V of all polynominals of degree >=3, togehter iwth 0, operations of P (P the set of polynomials)

now all the scalar multiplication axioms hold.
the text however says that the axion
$$\mbox{For u,v} \in V, \mbox{then} \ u+v \in V$$ does not hold

well ok take two polynomials
$$u(x) = a_{3} x^3 + ... + a_{n} x^n$$
$$v(x) = b_{3} x^3 + ... + b_{k} x^k$$
where both n,k>= 3, then suppose k< n
$$u(x) + v(x) = (a_{3} + b_{3}) x^3 + ... + (a_{k} + b_{k}) x^k + ... + a_{n} x^n$$
which is certainly a polynomial or degree >= 3 isnt it?
It also applies for n<k and n = k
is the text book wrong?

2. Feb 21, 2006

### matt grime

What is the degree of the following poly

x^4+1

now, do you see your error?

3. Feb 21, 2006

### stunner5000pt

ok let me correct that then
n,k >= 3
$$u(x) = a_{0} + a_{1} x + ... + a_{n} x^n$$
$$v(x) = b_{0} + b_{1} x + ... + b_{k} x^k$$
then for n< k
$$u + v = (a_{0} + b_{0}) + ... + (a_{k} + b_{k}) x^k + ... + a_{n} x^n$$

stil lseems to be of degree three to me
however if k=n and an= -bn then the polynomial is no more degree 3
is this corret?

4. Feb 21, 2006

### matt grime

Why don't you just find a counter example? two polys of degree 3 or greaterwhose sum isn't? A single counter examplem suffices.