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To decompose a vector in $\mathbb{R}^4$, you separate its components based on the bases of $W$ and $W^\perp$. The part corresponding to the basis of $W$ (the first two coordinates) forms a vector in $W$, and the part corresponding to the basis of $W^\perp$ (the last two coordinates) forms a vector in $W^\perp$.

Conditions: The ambient space is $\mathbb{R}^4$.; $W$ has basis $\left\{\begin{pmatrix}1\\0\\0\\0\end{pmatrix},\begin{pmatrix}0\\1\\0\\0\end{pmatrix}\right\}$.; $W^\perp$ has basis $\left\{\begin{pmatrix}0\\0\\1\\0\end{pmatrix},\begin{pmatrix}0\\0\\0\\1\end{pmatrix}\right\}$.; The target vector is given, e.g., $\begin{pmatrix}4\\3\\-2\\6\end{pmatrix}$.

To decompose a vector in $\mathbb{R}^4$, you separate its components based on the bases of $W$ and $W^\perp$. The part corresponding to the basis of $W$ (the first two coordinates) forms a vector in $W$, and the part corresponding to the basis of $W^\perp$ (the last two coordinates) forms a vector in $W^\perp$.

Conditions: The ambient space is $\mathbb{R}^4$.; $W$ has basis $\left\{\begin{pmatrix}1\\0\\0\\0\end{pmatrix},\begin{pmatrix}0\\1\\0\\0\end{pmatrix}\right\}$.; $W^\perp$ has basis $\left\{\begin{pmatrix}0\\0\\1\\0\end{pmatrix},\begin{pmatrix}0\\0\\0\\1\end{pmatrix}\right\}$.; The target vector is given, e.g., $\begin{pmatrix}4\\3\\-2\\6\end{pmatrix}$.