Find four vectors v1,v2,v3,v4
 in R^4
 such that
no two are collinear (in particular, none of the vectors is zero)

the set {v1,v2,v3,v4}
 is linearly dependent, and v4

 is not in Span{v1,v2,v3}

To find four vectors \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\) in \(\mathbb{R}^4\) that satisfy the following conditions:

1. **No two vectors are collinear** (i.e., none of the vectors is a scalar multiple of another, and none of the vectors is zero).
2. **The set \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\) is linearly dependent.**
3. **\(\mathbf{v}_4\) is not in \(\text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\)** (i.e., \(\mathbf{v}_4\) cannot be expressed as a linear combination of \(\mathbf{v}_1, \mathbf{v}_2,\) and \(\mathbf{v}_3\)).

We can construct the vectors as follows:

\[
\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \quad
\mathbf{v}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \quad
\mathbf{v}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \quad
\mathbf{v}_4 = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}.
\]

### Verification:

1. **No two vectors are collinear**:  
   - None of the vectors \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3,\) and \(\mathbf{v}_4\) are scalar multiples of each other.
   - Therefore, no two are collinear.

2. **The set \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\) is linearly dependent**:  
   - Consider the linear combination \(\mathbf{v}_1 + \mathbf{v}_2 + \mathbf{v}_3 - \mathbf{v}_4 = \mathbf{0}\).
   - Since the coefficients are not all zero, the set is linearly dependent.

3. **\(\mathbf{v}_4\) is not in \(\text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\)**:  
   - The span of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) includes all vectors of the form \(\begin{pmatrix} a \\ b \\ c \\ 0 \end{pmatrix}\) where \(a, b, c\) are real numbers.
   - Since \(\mathbf{v}_4 = \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}\), its last component (1) is nonzero, implying \(\mathbf{v}_4\) cannot be expressed as a linear combination of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\), which means it is not in the span of \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\).

These vectors satisfy all the given conditions.

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

1. What does it mean for a set of vectors to be linearly independent or dependent?
2. How would the conditions change if we wanted a linearly independent set?
3. What are the geometric interpretations of vectors being collinear or linearly dependent in \(\mathbb{R}^4\)?
4. Can you find another set of four vectors in \(\mathbb{R}^4\) that satisfy these conditions?
5. How does the concept of span relate to the dimension of a vector space?

**Tip:** When checking if a set of vectors is linearly dependent, consider forming a matrix with these vectors as columns and finding the determinant. If the determinant is zero, the set is linearly dependent.

Find four vectors v1, v2, v3, v4 in R^4 such that no two are collinear (in particular, none of the vectors is zero), the set {v1, v2, v3, v4} is linearly dependent, and v4 is not in Span{v1, v2, v3}.

Discover how to find four vectors in R^4 that are not collinear, form a linearly dependent set, and satisfy the condition that the fourth vector is not in the span of the first three. This problem highlights key concepts in linear algebra, including vector spaces and linear dependence.

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