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Denote the columns of the matrix 𝐴 by 𝑣1, 𝑣2, and 𝑣3. If 𝑏 is a linear combination of the columns of 𝐴, specify a linear combination of the columns of 𝐴 that equals 𝑏. If 𝑏 is not a linear combination of the columns of 𝐴, leave the spaces blank.

𝑣1+  𝑣2+ 𝑣3=𝑏.

Let's denote the columns of the matrix \( A \) by \( \mathbf{v}_1 \), \( \mathbf{v}_2 \), and \( \mathbf{v}_3 \):

\[
\mathbf{v}_1 = \begin{bmatrix} 3 \\ -1 \\ -2 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} -3 \\ -2 \\ 1 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 0 \\ -3 \\ 2 \end{bmatrix}
\]

Given that \( \mathbf{b} = \begin{bmatrix} 12 \\ -7 \\ -3 \end{bmatrix} \), we need to check if \( \mathbf{b} \) can be expressed as a linear combination of the columns of \( A \). This can be represented as:

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

This translates into solving the following system of linear equations:

\[
\begin{aligned}
3c_1 - 3c_2 + 0c_3 &= 12, \\
-1c_1 - 2c_2 - 3c_3 &= -7, \\
-2c_1 + 1c_2 + 2c_3 &= -3.
\end{aligned}
\]

I'll solve this system to determine the coefficients \( c_1 \), \( c_2 \), and \( c_3 \).

Let's solve the system.The solution to the system is:

\[
c_1 = 3, \quad c_2 = -1, \quad c_3 = 2
\]

So, \( \mathbf{b} \) can indeed be expressed as a linear combination of the columns of \( A \):

\[
\mathbf{b} = 3\mathbf{v}_1 - \mathbf{v}_2 + 2\mathbf{v}_3
\]

This is the linear combination you are looking for:

\[
3\mathbf{v}_1 - \mathbf{v}_2 + 2\mathbf{v}_3 = \mathbf{b}
\]

Would you like more details or have any further questions? Here are some related questions you might consider:

1. How do we interpret the coefficients in the linear combination?
2. What does it mean if \( \mathbf{b} \) cannot be expressed as a linear combination of the columns of \( A \)?
3. How can we verify if a given vector is in the span of the columns of a matrix?
4. How do these concepts relate to the idea of linear independence?
5. What is the geometric interpretation of the linear combination in three-dimensional space?

**Tip:** Always check the rank of the matrix \( A \) to determine the dimension of the space spanned by its columns.

Explore how to determine if vector b can be expressed as a linear combination of the columns of matrix A. Learn the process of finding coefficients using matrix operations and linear algebra concepts. Suitable for college-level students and those interested in linear algebra.

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