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}_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.