In this image, we are given a basis $C$ of $\mathbb{R}^n$ and a vector $[\vec{x}]_C$ with its coordinates relative to the basis $C$. Our task is likely to find the vector $\vec{x}$ in the standard basis of $\mathbb{R}^n$ using the given information.

Information given:

• Basis $C = \left\{ \begin{pmatrix} 2 \\ 2 \\ 0 \\ -1 \end{pmatrix}, \begin{pmatrix} -1 \\ -3 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -3 \\ 2 \\ 0 \\ 0 \end{pmatrix} \right\}$ is a set of four vectors in $\mathbb{R}^n$.
• The vector $[\vec{x}]_C = \begin{pmatrix} 1 \\ -4 \\ -3 \\ -2 \end{pmatrix}$ represents the coordinates of $\vec{x}$ in the basis $C$.

To find:

We are tasked with finding the vector $\vec{x}$ in the standard basis. We use the relationship: $\vec{x} = C [\vec{x}]_C$ where $C$ is the matrix whose columns are the basis vectors, and $[\vec{x}]_C$ is the coordinate vector in this basis.

Let me write down the necessary steps:

1. Construct the matrix $C$ from the basis vectors.