Let $S = \mathbb{R}^n$ be a subspace with dimension $n$. Also, let $\{\phi_j\}_{j=1}^k$ be $k$ orthonormal vector that describes another subspace $\Psi_k \subseteq \mathbb{R}^n$ with dimension $k$ where each $\phi_j \in \mathbb{R}^n$ and $k \leq n$.
Can we write the following?
$$ S=\Psi_k \bigoplus \Psi_k^{\perp} $$
where $\Psi_k^{\perp}$ is orthogonal complement of $\Psi_k$ and has another orthonormal set $\{\phi_j\}_{j=k+1}^n$ as its basis.
Let $y$ be an arbitrary vector in $\mathbb{R}^n$.
Can we write the following?
$$ y = \langle \phi_1,y \rangle \phi_1 + \langle \phi_2,y \rangle \phi_2 + \cdots + \langle \phi_n,y \rangle \phi_n $$
$(1)$ Yes. Every subspace in finite-dimensions is closed, so we have the decomposition $\mathbb R^n=\Psi_k\oplus\Psi_k^\perp$ by the orthogonal decomposition theorem.
$(2)$ Yes, you can write $y=\langle\phi_1,y\rangle \phi_1+\cdots+\langle\phi_n,y\rangle\phi_n$. This is a property of orthonormal bases, in that you can write any vector as a linear combination of your orthonormal basis, and the coefficients will be these inner products.