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We are given two vectors \(\mathbf{v} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\) and \(\mathbf{w} = \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\) in \(\mathbb{R}^3\), and we are asked to find a vector that **cannot** be written as a linear combination of \(\mathbf{v}\) and \(\mathbf{w}\).

### Step-by-Step Solution:

A linear combination of \(\mathbf{v}\) and \(\mathbf{w}\) would take the form:

\[
\mathbf{c} = a\mathbf{v} + b\mathbf{w} = a \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + b \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} a \\ b \\ a + b \end{pmatrix}
\]

Thus, any vector \(\mathbf{c}\) that can be written as a linear combination of \(\mathbf{v}\) and \(\mathbf{w}\) must have the form:

\[
\mathbf{c} = \begin{pmatrix} a \\ b \\ a + b \end{pmatrix}
\]

### Condition for a vector not to be in the span:
For a vector \(\mathbf{u} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\) to **not** be a linear combination of \(\mathbf{v}\) and \(\mathbf{w}\), it must **not** satisfy the relation \( z = a + b \), which is equivalent to \( z = x + y \) since \( a = x \) and \( b = y \).

### Example:
Consider the vector \(\mathbf{u} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\). We check if it satisfies the relation \( z = x + y \):

\[
0 \neq 1 + 1
\]

Hence, \(\mathbf{u} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\) does **not** lie in the span of \(\mathbf{v}\) and \(\mathbf{w}\), because it violates the condition \( z = x + y \).

### Conclusion:
The vector \(\mathbf{u} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\) cannot be written as a linear combination of \(\mathbf{v}\) and \(\mathbf{w}\).

Would you like further details on this solution or explanations on linear combinations?

Here are 5 related questions:
1. How do you find the span of a set of vectors?
2. What is the geometric interpretation of the span of two vectors in \(\mathbb{R}^3\)?
3. How do you determine if a set of vectors is linearly independent?
4. How can we extend this concept to higher dimensions (\(\mathbb{R}^n\))?
5. What are the properties of a vector space spanned by two vectors?

**Tip**: To check if a vector belongs to the span of two other vectors, solve for the coefficients in the linear combination equation. If no solution exists, the vector is not in the span.

Consider the two vectors v = [1, 0, 1] and w = [0, 1, 1] in R^3. Find a vector in R^3 that cannot be written as a linear combination of v and w. Justify your answer.

Learn how to find a vector in R^3 that cannot be expressed as a linear combination of two given vectors v = [1, 0, 1] and w = [0, 1, 1]. This solution explores concepts in linear algebra such as vector spaces, spans, and linear independence, suitable for undergraduate students.

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