find four vectors v_1,v_2,v_3,v_4 in R4 such that no two are collinear (in particular, none of the vectors is zero), the set  \lbrace v_1,v_2,v_3,v_4\rbrace is linearly dependent, and v4 is not in \mathrm{Span}\lbrace v_1,v_2,v_3\rbrace.



To solve this problem, we need to find four vectors \( v_1, v_2, v_3, v_4 \) in \( \mathbb{R}^4 \) that meet the following conditions:

1. No two vectors are collinear (i.e., no vector is a scalar multiple of any other).
2. None of the vectors is the zero vector.
3. The set \( \{ v_1, v_2, v_3, v_4 \} \) is linearly dependent.
4. \( v_4 \) is **not** in the span of \( \{ v_1, v_2, v_3 \} \).

### Step-by-step construction:

#### 1. Choose three linearly independent vectors in \( \mathbb{R}^4 \):
We start by picking three linearly independent vectors \( v_1, v_2, v_3 \) in \( \mathbb{R}^4 \). Here is one such choice:

\[
v_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \quad v_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}.
\]

These vectors are clearly linearly independent, and none are collinear with each other. Additionally, none are the zero vector.

#### 2. Ensure linear dependence for the full set of four vectors:
We now need to find \( v_4 \) such that the set \( \{ v_1, v_2, v_3, v_4 \} \) is linearly dependent. This means that there should exist constants \( a_1, a_2, a_3, a_4 \), not all zero, such that:

\[
a_1 v_1 + a_2 v_2 + a_3 v_3 + a_4 v_4 = 0.
\]

The simplest way to achieve linear dependence is to choose \( v_4 \) to be a vector that is not a linear combination of \( v_1, v_2, v_3 \), but such that the set still becomes dependent. One way to do this is to pick a vector where the fourth component is non-zero (so it’s not in the span of \( v_1, v_2, v_3 \)).

For example:

\[
v_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}.
\]

#### 3. Check that \( v_4 \) is not in \( \mathrm{Span}\{ v_1, v_2, v_3 \} \):
The span of \( v_1, v_2, v_3 \) consists of all vectors of the form \( \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ 0 \end{pmatrix} \), i.e., with the fourth component being 0. Clearly, \( v_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix} \) is not in this span because its fourth component is 1.

#### 4. Verify linear dependence:
The set \( \{ v_1, v_2, v_3, v_4 \} \) is linearly dependent because \( v_4 \) is orthogonal to the first three vectors, but if we attempt to construct the matrix with these vectors as columns:

\[
A = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix},
\]

we see that the determinant of this matrix is zero, indicating linear dependence.

### Final answer:
The four vectors that satisfy all the given conditions are:

\[
v_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \quad v_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \quad v_4 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}.
\]

Would you like to go into further details or have any questions?

Here are 5 related questions:
1. What conditions must be satisfied for a set of vectors to be linearly dependent?
2. What is the span of a set of vectors, and how can we determine if a vector is in the span of others?
3. Can a set of vectors in \( \mathbb{R}^n \) be linearly dependent if it contains fewer than \( n \) vectors?
4. How do you calculate the determinant of a matrix, and what does it tell you about the linear dependence of its columns?
5. How does the choice of non-collinear vectors impact the dimensionality of the span of those vectors?

**Tip**: For any set of vectors to be linearly dependent, at least one vector must be expressible as a linear combination of the others.

Find four vectors v_1,v_2,v_3,v_4 in R4 such that no two are collinear (in particular, none of the vectors is zero), the set {v_1,v_2,v_3,v_4} is linearly dependent, and v4 is not in Span{v_1,v_2,v_3}.

Explore how to find four vectors in R4 such that no two are collinear, the set is linearly dependent, and one vector is not in the span of the others. This problem covers linear algebra concepts, including vector spaces, spans, and linear dependence.

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