Four non zero vectors will always be a. linearly dependent b. linearly independent c. either a or b d. none of these

Options

A. linearly dependent
B. linearly independent
C. either a or b
D. none of these

Explanation

To solve the question regarding the linear dependence or independence of four non-zero vectors, we can follow these steps:

### Step 1: Understand the Definitions
First, we need to understand what linear dependence and independence mean:
- **Linearly Independent Vectors**: A set of vectors is said to be linearly independent if the only solution to the equation \(c_1\mathbf{v_1} + c_2\mathbf{v_2} + c_3\mathbf{v_3} + c_4\mathbf{v_4} = \mathbf{0}\) is \(c_1 = c_2 = c_3 = c_4 = 0\).
- **Linearly Dependent Vectors**: A set of vectors is linearly dependent if there exists a non-trivial solution (not all coefficients are zero) to the equation above.

### Step 2: Apply the Concept of Dimensions
In a vector space, the maximum number of linearly independent vectors is equal to the dimension of that space. For example:
- In 2D space, at most 2 vectors can be linearly independent.
- In 3D space, at most 3 vectors can be linearly independent.

### Step 3: Analyze the Given Vectors
Given that we have four non-zero vectors, we need to consider the dimensionality:
- In a 3D space, we can only have up to 3 linearly independent vectors. Therefore, if we have four vectors in 3D space, at least one of them must be expressible as a linear combination of the others, which means they are linearly dependent.

### Step 4: Conclusion
Since four non-zero vectors cannot all be linearly independent in a space of dimension 3 or lower, we conclude that they must be linearly dependent.

### Final Answer
The correct option is:
**a. linearly dependent**

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