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Polynomial of degree $鈮�7$ with ${\mathrm{a}}_{\mathrm{odd}}=0$.

LetVbe a non-empty set and addition is an internal binary operation which maps $\mathbf{V}\mathbf{脳}\mathbf{V}\mathbf{鈫�}\mathbf{V}$ which assigns to each order pair (x,y) $\mathbf{鈭�}\mathbf{V}\mathbf{脳}\mathbf{V}$to a unique element of Vdenoted by x+yand multiplication is an external binary operation over the field Fthat maps $\mathbf{F}\mathbf{脳}\mathbf{V}\mathbf{鈫�}\mathbf{V}$ which assigns to each order pair $\mathbf{(}\mathbf{伪}\mathbf{,}\mathbf{x}\mathbf{)}\mathbf{鈭�}\mathbf{F}\mathbf{脳}\mathbf{V}$to a unique element in V denoted by $\mathbf{伪}\mathbf{.}\mathbf{x}\mathbf{=}\mathbf{伪虫}$ , then is called the vector space if it satisfies the following conditions:

(V,+)is an Abelian group

$$\begin{gathered} \mathbf{伪}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{=}\mathbf{伪虫}\mathbf{+}\mathbf{伪测}\mathbf{鈭赌}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{鈭�}\mathbf{V}\mathbf{}\mathbf{and}\mathbf{}\mathbf{伪}\mathbf{鈭�}\mathbf{F} \\ \mathbf{(}\mathbf{伪}\mathbf{+}\mathbf{尾}\mathbf{)}\mathbf{x}\mathbf{=}\mathbf{伪虫}\mathbf{+}\mathbf{尾虫}\mathbf{鈭赌}\mathbf{x}\mathbf{鈭�}\mathbf{V}\mathbf{}\mathbf{and}\mathbf{}\mathbf{伪}\mathbf{,}\mathbf{尾}\mathbf{鈭�}\mathbf{F} \\ \mathbf{伪}\mathbf{\left(}\mathbf{尾虫}\mathbf{\right)}\mathbf{=}\mathbf{\left(}\mathbf{伪尾}\mathbf{\right)}\mathbf{x}\mathbf{=}\mathbf{尾}\mathbf{\left(}\mathbf{伪虫}\mathbf{\right)}\mathbf{鈭赌}\mathbf{伪}\mathbf{,}\mathbf{尾}\mathbf{鈭�}\mathbf{F}\mathbf{}\mathbf{and}\mathbf{}\mathbf{x}\mathbf{鈭�}\mathbf{V} \\ \mathbf{1}\mathbf{.}\mathbf{x}\mathbf{=}\mathbf{x}\mathbf{鈭赌}\mathbf{x}\mathbf{鈭�}\mathbf{V} \end{gathered}$$

As Polynomial of degree $鈮�7$ is given so this polynomial can be written as,

$\mathrm{P}\left(\mathrm{x}\right)={\mathrm{a}}_0+{\mathrm{a}}_1\mathrm{x}+{\mathrm{a}}_2{\mathrm{x}}^2+{\mathrm{a}}_3{\mathrm{x}}^3+{\mathrm{a}}_4{\mathrm{x}}^4+{\mathrm{a}}_5{\mathrm{x}}^5+{\mathrm{a}}_6{\mathrm{x}}^6+{\mathrm{a}}_7{\mathrm{x}}^7$

If ${\mathrm{a}}_{\mathrm{odd}}=0$, then

$\mathrm{P}\left(\mathrm{x}\right)={\mathrm{a}}_0+{\mathrm{a}}_2{\mathrm{x}}^2+{\mathrm{a}}_4{\mathrm{x}}^4+{\mathrm{a}}_6{\mathrm{x}}^6$

From the example 1,

Add two polynomials of this form will give another polynomial of the same form.

Addition of algebraic expression is commutative and associative.

The "zero vector鈥� is the polynomial with all coefficient ${\mathrm{a}}_{\mathrm{i}}=0$ and adding it to any other polynomial just gives that other polynomial. Also the additive inverse of a function $\mathrm{f}\left(\mathrm{x}\right)\mathrm{is}-\mathrm{f}\left(\mathrm{x}\right)$and $-\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)=0$as required for a vector space.

Hence all the properties of vector space are satisfied so, this set is a vector space.

Now the functions $\left\{1,{\mathrm{x}}^2,{\mathrm{x}}^4,{\mathrm{x}}^6\right\}$span the vector space i.e. any element of the vector space can be written as a linear combination of them. This show that they are linearly independent.

Therefore, the functions $\left\{1,{\mathrm{x}}^2,{\mathrm{x}}^4,{\mathrm{x}}^6\right\}$ form the basis for the vector space, and therefore the vector space has dimension 4.