Consider the following.
x =

4

0
−4
, basis B =

1

0
0
,

0

1
0
,

0

0
1

, basis C =

1

1
1
,

0

1
1
,

0

0
1

in
ℝ3
(a)
Find the coordinate vectors
[x]

basis B
and
[x]

basis C
of x with respect to the bases basis B and basis C, respectively.

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:
\[
\begin{cases}
c_1 = 4 \\
c_1 + c_2 = 0 \\
c_1 + c_2 + c_3 = -4 \\
\end{cases}
\]

Solving these equations step-by-step:
1. From the first equation: \( c_1 = 4 \)
2. Substitute \( c_1 = 4 \) into the second equation: \( 4 + c_2 = 0 \Rightarrow c_2 = -4 \)
3. Substitute \( c_1 = 4 \) and \( c_2 = -4 \) into the third equation: \( 4 - 4 + c_3 = -4 \Rightarrow c_3 = -4 \)

So, the coordinate vector \([x]_C\) is:
\[ [x]_C = \begin{pmatrix} 4 \\ -4 \\ -4 \end{pmatrix} \]

### Summary
1. The coordinate vector \([x]_B\) with respect to basis \(B\) is:
\[ [x]_B = \begin{pmatrix} 4 \\ 0 \\ -4 \end{pmatrix} \]

2. The coordinate vector \([x]_C\) with respect to basis \(C\) is:
\[ [x]_C = \begin{pmatrix} 4 \\ -4 \\ -4 \end{pmatrix} \]

Would you like details or have any questions?

Here are 8 related questions that expand on this information:
1. How do you find the coordinate vector of a point with respect to a given basis?
2. What is the standard basis in \(\mathbb{R}^3\)?
3. How do you solve a system of linear equations to find the coordinates in a different basis?
4. What is the geometric interpretation of changing bases in \(\mathbb{R}^3\)?
5. How do you verify if a set of vectors forms a basis for \(\mathbb{R}^3\)?
6. What are the properties of a basis in vector space?
7. How can you transform coordinates from one basis to another?
8. What is the significance of the coordinate vectors in applications such as computer graphics?

**Tip:** Always ensure the new basis vectors are linearly independent to form a valid basis for the vector space.

Learn how to find coordinate vectors [x]_B and [x]_C of a vector x in ℝ³ with respect to different bases B and C. Includes step-by-step solutions and explanations for vector space transformations. Suitable for advanced undergraduate students and anyone interested in linear algebra concepts.

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