show that any set of 4 vectors in R-2 is linearly dependent

In \( \mathbb{R}^2 \), a vector has two components, and thus the space has dimension 2. This means any set of more than 2 vectors in \( \mathbb{R}^2 \) is guaranteed to be linearly dependent. Here's a step-by-step explanation for why any set of 4 vectors in \( \mathbb{R}^2 \) must be linearly dependent:

### 1. **Linear Independence Definition**:
   A set of vectors \( \{v_1, v_2, \dots, v_k\} \) is **linearly independent** if the only solution to the equation
   \[
   c_1 v_1 + c_2 v_2 + \dots + c_k v_k = 0
   \]
   is \( c_1 = c_2 = \dots = c_k = 0 \) (i.e., the only way to express the zero vector as a linear combination of these vectors is by having all the coefficients equal to zero).

   If there exists a non-trivial solution (where at least one \( c_i \neq 0 \)), then the vectors are **linearly dependent**.

### 2. **The Dimension of \( \mathbb{R}^2 \)**:
   The space \( \mathbb{R}^2 \) has dimension 2. This means any set of more than 2 vectors in \( \mathbb{R}^2 \) will span a space that can be described by at most 2 linearly independent vectors.

### 3. **Applying to 4 Vectors**:
   - Suppose we have 4 vectors in \( \mathbb{R}^2 \), say \( \{v_1, v_2, v_3, v_4\} \).
   - Since \( \mathbb{R}^2 \) is a 2-dimensional space, any set of more than 2 vectors must be linearly dependent.
   - This is because, in a 2-dimensional space, at most 2 vectors can be linearly independent. Adding a third vector must result in it being expressible as a linear combination of the first two.

### 4. **Mathematical Proof**:
   Consider 4 vectors \( v_1, v_2, v_3, v_4 \in \mathbb{R}^2 \). Each vector can be written as:
   \[
   v_i = \begin{pmatrix} a_i \\ b_i \end{pmatrix}, \quad i = 1, 2, 3, 4.
   \]
   Now, we need to determine if the equation
   \[
   c_1 v_1 + c_2 v_2 + c_3 v_3 + c_4 v_4 = 0
   \]
   has a non-trivial solution. This expands to:
   \[
   c_1 \begin{pmatrix} a_1 \\ b_1 \end{pmatrix} + c_2 \begin{pmatrix} a_2 \\ b_2 \end{pmatrix} + c_3 \begin{pmatrix} a_3 \\ b_3 \end{pmatrix} + c_4 \begin{pmatrix} a_4 \\ b_4 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.
   \]
   Writing this out in terms of components gives:
   \[
   c_1 a_1 + c_2 a_2 + c_3 a_3 + c_4 a_4 = 0 \quad \text{(for the first component)},
   \]
   \[
   c_1 b_1 + c_2 b_2 + c_3 b_3 + c_4 b_4 = 0 \quad \text{(for the second component)}.
   \]
   This is a system of 2 equations with 4 unknowns (\(c_1, c_2, c_3, c_4\)). In general, such a system has infinitely many solutions (since there are more unknowns than equations), implying that at least one \( c_i \neq 0 \), meaning the vectors are linearly dependent.

### Conclusion:
Any set of 4 vectors in \( \mathbb{R}^2 \) must be linearly dependent because the dimension of \( \mathbb{R}^2 \) is 2, and you cannot have more than 2 linearly independent vectors in a 2-dimensional space.

Would you like more details on any part of this explanation or any follow-up questions?

### Related Questions:
1. What is the definition of linear dependence and independence in \( \mathbb{R}^n \)?
2. How can we extend this result to \( \mathbb{R}^n \) for any \( n \)?
3. What is the geometric interpretation of linearly dependent vectors in \( \mathbb{R}^2 \)?
4. Can we have 3 linearly independent vectors in \( \mathbb{R}^3 \)?
5. How do you determine if a set of vectors is linearly dependent using a matrix?

### Tip:
To check if a set of vectors is linearly dependent, you can form a matrix with these vectors as columns. If the matrix has a determinant of zero or its rank is less than the number of vectors, the vectors are linearly dependent.

Explore why any set of 4 vectors in R^2 must be linearly dependent using linear algebra concepts. This explanation covers the linear dependence theorem, formulas, and step-by-step reasoning, suitable for undergraduate students studying vector spaces and dimensionality.

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