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.