https://file.aiquickdraw.com/mathbot/file/1728002157856gmosk1ls.jpg

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.

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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).

Write the vector [-1, 1, -3] as a linear combination of the vectors [1, 1, 2], [1, 2, 0], and [2, 3, 1]. Also, find a vector that is not in the span of these vectors.

Learn how to express a vector as a linear combination of three given vectors and find a vector outside their span. This problem involves key concepts from linear algebra such as vector spaces and spans, suitable for undergraduate-level studies.

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