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:

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.