To find the coordinate vectors $[x]_B$ and $[x]_C$ of $x$ with respect to the bases $B$ and $C$, we need to express $x$ in terms of each basis.

Given

$x = \begin{pmatrix} 4 \\ 0 \\ -4 \end{pmatrix}$

Basis $B$:

$B = \left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right\}$

Basis $C$:

$C = \left\{ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right\}$

Step-by-Step Solution

(a) Finding $[x]_B$:

Since basis $B$ is the standard basis of $\mathbb{R}^3$, the coordinates of $x$ in this basis are simply the components of $x$. Thus, $[x]_B = \begin{pmatrix} 4 \\ 0 \\ -4 \end{pmatrix}$

(b) Finding $[x]_C$:

To find $[x]_C$, we need to solve the equation: $x = c_1 \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + c_2 \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} + c_3 \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$

This can be written as a system of linear equations: $\begin{pmatrix} 4 \\ 0 \\ -4 \end{pmatrix} = c_1 \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} + c_2 \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} + c_3 \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$

Breaking it down into equations: