To find the coordinates of the vector $\hat{u}$ relative to the given basis $\mathcal{B}$ in each problem, we need to express $\hat{u}$ as a linear combination of the basis vectors. Specifically, if the basis is $\mathcal{B} = \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \}$, we want to find scalars $c_1$, $c_2$, and $c_3$ such that:

$\hat{u} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3 \mathbf{v}_3$

We can set this up as a system of equations or a matrix equation to solve for the scalars $c_1$, $c_2$, and $c_3$.

Problem 3

1. Basis vectors: $\mathbf{v}_1 = \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} -2 \\ 1 \\ 4 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} 8 \\ 1 \\ 3 \end{pmatrix}$

2. Vector $\hat{u}$: $\hat{u} = \begin{pmatrix} 2 \\ 2 \\ 5 \end{pmatrix}$

Set up the matrix equation: $\begin{pmatrix} 0 & -2 & 8 \\ 3 & 1 & 1 \\ 1 & 4 & 3 \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \\ 5 \end{pmatrix}$

Solving this system will give the coordinates of $\hat{u}$ relative to the basis $\mathcal{B}$.

Problem 4

1. Basis vectors: $\mathbf{v}_1 = \begin{pmatrix} 4 \\ 7 \\ -2 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} -4 \\ 1 \\ 6 \end{pmatrix}$

2. Vector $\hat{u}$: $\hat{u} = \begin{pmatrix} -1 \\ 0 \\ 3 \end{pmatrix}$

Set up the matrix equation: $\begin{pmatrix} 4 & 0 & -4 \\ 7 & 0 & 1 \\ -2 & 1 & 6 \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix} = \begin{pmatrix} -1 \\ 0 \\ 3 \end{pmatrix}$

Solving this system will yield the coordinates of $\hat{u}$ relative to the basis $\mathcal{B}$ for Problem 4.

Let me proceed with solving these systems to find the coordinates for both problems.The coordinates of the vector $\hat{u}$ relative to the given basis for each problem are as follows:

Problem 3

$\hat{u}_{\mathcal{B}} = \begin{pmatrix} 0.2308 \\ 0.8462 \\ 0.4615 \end{pmatrix}$

Problem 4

$\hat{u}_{\mathcal{B}} = \begin{pmatrix} -0.0313 \\ 1.6250 \\ 0.2188 \end{pmatrix}$

Would you like more details on how these calculations were performed, or do you have any questions?

Here are some related questions you might find interesting:

1. How do we determine if a set of vectors forms a basis?
2. What is the geometric interpretation of coordinates relative to a basis?
3. Can these coordinates change if we choose a different basis?
4. How do we verify if the calculated coordinates are correct?
5. What are the applications of basis transformations in linear algebra?

Tip: Understanding the matrix equation setup for changing coordinates is essential for handling more complex transformations in linear algebra.