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To determine if the vector \(\mathbf{v} = \begin{bmatrix} 2 \\ -1 \\ 3 \\ 0 \end{bmatrix}\) is a linear combination of the vectors \(\mathbf{v}_1 = \begin{bmatrix} 1 \\ 3 \\ 2 \\ 0 \end{bmatrix}\), \(\mathbf{v}_2 = \begin{bmatrix} 5 \\ 1 \\ 2 \\ 4 \end{bmatrix}\), and \(\mathbf{v}_3 = \begin{bmatrix} -1 \\ 0 \\ 2 \\ 1 \end{bmatrix}\), we need to solve the following system of linear equations:

\[
c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3 \mathbf{v}_3 = \mathbf{v}
\]

This translates to the matrix equation:

\[
c_1 \begin{bmatrix} 1 \\ 3 \\ 2 \\ 0 \end{bmatrix} + c_2 \begin{bmatrix} 5 \\ 1 \\ 2 \\ 4 \end{bmatrix} + c_3 \begin{bmatrix} -1 \\ 0 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ -1 \\ 3 \\ 0 \end{bmatrix}
\]

Or, more explicitly:

\[
\begin{bmatrix} 1 & 5 & -1 \\ 3 & 1 & 0 \\ 2 & 2 & 2 \\ 0 & 4 & 1 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 2 \\ -1 \\ 3 \\ 0 \end{bmatrix}
\]

Now, we solve this system using either Gaussian elimination or matrix inversion.

I'll proceed with solving it for you.The solution to the system of equations is approximately:

\[
c_1 \approx -0.0067, \quad c_2 \approx 0.2937, \quad c_3 \approx 0.5232
\]

However, the system has a non-zero residual of about 7.54, which indicates that the vector \(\mathbf{v} = \begin{bmatrix} 2 \\ -1 \\ 3 \\ 0 \end{bmatrix}\) is **not** an exact linear combination of \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\).

Would you like a deeper explanation of these results or further details?

Here are some related questions you might explore:
1. What does it mean for a vector to be a linear combination of others?
2. How does Gaussian elimination work for solving linear systems?
3. What is the significance of the residual in a least squares solution?
4. Can we always solve any system of linear equations?
5. How does matrix rank affect the solvability of systems?

**Tip**: Non-zero residuals in least squares indicate the system is overdetermined or inconsistent.

In R^4, is v = [2, -1, 3, 0] a linear combination of v1 = [1, 3, 2, 0], v2 = [5, 1, 2, 4], and v3 = [-1, 0, 2, 1]?

Explore the solution to determine if the vector v = [2, -1, 3, 0] is a linear combination of the vectors v1 = [1, 3, 2, 0], v2 = [5, 1, 2, 4], and v3 = [-1, 0, 2, 1]. This solution involves solving a system of linear equations using Gaussian elimination in R^4.

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