show that linear space is direct sum of two subspaces and find projections

Question

I have never seen this task type so do not know how to even start. have no thoughts, please send me right direction

I have to show that linear space $\mathbb{R_4}[x]$ is direct sum of two subspaces and also find projection of $g(x)$ to $L_1 ||L_2$ and projection $L_2||L_1$:

$$L_1=\{f(x) \in \mathbb{R_4}[x]\mid f(1)=f''(1). \quad f(1)=f(-1)\} $$ $$L_2=\langle x^2+x,x^3+1 \rangle$$ and $$g(x) =x^4+3x+4$$

Answer 1

To show it's a direct sum you have to check $L_1$ has dimension $3$ and $L_2$ has dimension $2$, and that $L_1\cap L_2=\{0\}$.

For the projection of $g(x)$ onto $L_1$ parallel to $L_2$, you need to find coefficients $\lambda, mu$ such that $\;g(x)-\lambda(x^2+x)-\mu(x^3+1)$ satisfies the conditions defining $g$.

The projection of $g(x)$ onto $L_2$ parallel to $L_1$ is of course $\lambda(x^2+x)+\mu(x^3+1)$.