Jtdylktuy

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 $mathbb R^n=Psi_koplusPsi_k^perp$ by the orthogonal decomposition theorem.

$(2)$ Yes, you can write $y=langlephi_1,yrangle phi_1+cdots+langlephi_n,yranglephi_n$. 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.