Let's break down the two parts of the question from the image:

(c) Linear Combination:

You are asked to write the vector $\begin{bmatrix} -1 \\ 1 \\ -3 \end{bmatrix}$ as a linear combination of three given vectors: $\mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}.$ This means solving for scalars $a$, $b$, and $c$ such that: $a \mathbf{v}_1 + b \mathbf{v}_2 + c \mathbf{v}_3 = \begin{bmatrix} -1 \\ 1 \\ -3 \end{bmatrix}.$ This translates into the system of equations: $a(1) + b(1) + c(2) = -1$ $a(1) + b(2) + c(3) = 1$ $a(2) + b(0) + c(1) = -3$ We can solve this system to find $a$, $b$, and $c$.

(d) Find a Vector Not in the Span:

You need to find a vector that is not in the span of the three given vectors: $\mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}.$ The span of these vectors is the set of all possible linear combinations of them. To find a vector that is not in this span, one approach is to check whether a given vector satisfies the condition of being a linear combination of $\mathbf{v}_1$, $\mathbf{v}_2$, and $\mathbf{v}_3$. If it doesn't, then it lies outside the span.

Do you want me to solve these systems and find a vector outside the span?

Here are some follow-up questions:

1. What does it mean for vectors to be linearly independent?
2. How do you determine the dimension of the span of a set of vectors?
3. What is the geometric interpretation of the span of vectors in $\mathbb{R}^3$?
4. What methods can be used to solve systems of linear equations (e.g., substitution, Gaussian elimination)?
5. How does a basis relate to the concept of span?

Tip: When working with spans and vector spaces, always ensure that the number of vectors is appropriate for the dimension you're working in (e.g., three vectors should span at most a 3D space).