Show that the vector of x1=[q,2,3] x2=[3,-2,1]    x3=[1,-6-5] x4=[4374] are linear dependence 

To prove that the vectors \( \mathbf{x}_1 = [q, 2, 3] \), \( \mathbf{x}_2 = [3, -2, 1] \), \( \mathbf{x}_3 = [1, -6, -5] \), and \( \mathbf{x}_4 = [4, 3, 7, 4] \) are linearly dependent, we need to check if there exist scalars \( c_1, c_2, c_3, c_4 \) (not all zero) such that:

\[
c_1 \mathbf{x}_1 + c_2 \mathbf{x}_2 + c_3 \mathbf{x}_3 + c_4 \mathbf{x}_4 = \mathbf{0}
\]

However, there's a mismatch in the number of components. The first three vectors \( \mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3 \) are 3-dimensional, while the fourth vector \( \mathbf{x}_4 \) is 4-dimensional. Since vectors must have the same number of components to be added together or compared, these four vectors cannot be combined to form a system of linear dependence.

Would you like to clarify the question, perhaps correcting the dimensions of the vectors, or focus on a subset of these vectors?

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Here are 5 related questions to extend the topic:

1. How do we check for linear dependence for vectors of equal dimension?
2. What does it mean if a set of vectors is linearly independent?
3. Can vectors of different dimensions ever be compared in a system of equations?
4. What role does the rank of a matrix play in determining linear dependence?
5. How can Gaussian elimination be used to check for linear dependence?

**Tip:** Linear dependence occurs if one vector in a set can be expressed as a linear combination of the others.

Show that the vector of x1=[q,2,3] x2=[3,-2,1] x3=[1,-6-5] x4=[4374] are linear dependence

This example explains how to prove linear dependence for vectors x1=[q,2,3], x2=[3,-2,1], x3=[1,-6,-5], and x4=[4,3,7,4]. Understand the concepts of linear dependence, vector spaces, and how different dimensions affect vector comparisons.

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