To solve this problem, we need to express the vector v in terms of the ordered basis $\mathcal{B} = \{ \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3 \}$.

Problem summary:

We are given:

• The basis vectors: $\mathbf{u}_1 = \begin{bmatrix} 5 \\ 4 \\ -1 \end{bmatrix}, \quad \mathbf{u}_2 = \begin{bmatrix} 8 \\ 6 \\ 3 \end{bmatrix}, \quad \mathbf{u}_3 = \begin{bmatrix} 7 \\ 1 \\ -3 \end{bmatrix}$
• The vector v: $\mathbf{v} = \begin{bmatrix} -3 \\ -7 \\ -1 \end{bmatrix}$

We need to find the coordinates of v with respect to the basis $\mathcal{B}$, i.e., find $[\mathbf{v}]_{\mathcal{B}}$, which are the scalars $c_1, c_2, c_3$ such that: $\mathbf{v} = c_1 \mathbf{u}_1 + c_2 \mathbf{u}_2 + c_3 \mathbf{u}_3$

This is equivalent to solving the matrix equation: $\begin{bmatrix} \mathbf{u}_1 & \mathbf{u}_2 & \mathbf{u}_3 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \mathbf{v}$ Which can be written as: $\begin{bmatrix} 5 & 8 & 7 \\ 4 & 6 & 1 \\ -1 & 3 & -3 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} -3 \\ -7 \\ -1 \end{bmatrix}$

I'll solve this system of equations using matrix methods to find $c_1, c_2, c_3$.The coordinates of the vector $\mathbf{v}$ with respect to the basis $\mathcal{B}$ are approximately: $[\mathbf{v}]_{\mathcal{B}} = \begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}$

Thus, $c_1 = -2$, $c_2 \approx 0$, and $c_3 = 1$.

Would you like further details or explanations?

Here are some related questions for deeper understanding:

1. What are the properties of a basis in vector spaces?
2. How can you check if a set of vectors forms a basis?
3. What happens if the matrix of basis vectors is singular?
4. How can this problem be solved using other methods like Gaussian elimination?
5. How do coordinate transformations work between different bases?

Tip: Always verify the solution by multiplying the basis matrix by the found coordinates to check if it reconstructs the original vector.