Understanding Linear Dependence of Vectors: A Case Study

Linear dependence is a fundamental concept in linear algebra, where a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be expressed as a linear combination of the others. In this article, we will analyze the linear dependence of two specific vectors, and , through the lens of linear equations and matrix operations. This analysis will help us understand the concept of linear dependence more thoroughly and align with Google's SEO standards for rich, informative content.

Introduction to Linear Dependence

The concept of linear dependence is crucial in various applications, from computer graphics to data analysis. Two vectors are said to be linearly dependent if there exist scalars and , not both zero, such that the equation c_1mathbf{v_1} c_2mathbf{v_2} mathbf{0} holds true. Here, represents the zero vector.

Analysis of Vectors and

Let's investigate whether and are linearly dependent by forming the equation:

c_1mathbf{v_1} - c_2mathbf{v_2} mathbf{0}

This can be rewritten as:

c_1 begin{pmatrix} 2 3 1 -2 end{pmatrix} - c_2 begin{pmatrix} -6 -9 -3 6 end{pmatrix} begin{pmatrix} 0 0 0 0 end{pmatrix}

This leads to the following system of linear equations:

2 - 6 0 3 - 9 0 - 3 0 -2 6 0

Solving the System of Equations

From the first equation:

2c_1 6c_2 Rightarrow c_1 3c_2

Substituting into the third equation:

3c_2 - 3c_2 0

This holds true for any value of . Choosing gives:

c_1 3

Thus, we have a nontrivial solution:

3mathbf{v_1} - 1mathbf{v_2} 3begin{pmatrix} 2 3 1 -2 end{pmatrix} - 1begin{pmatrix} -6 -9 -3 6 end{pmatrix} begin{pmatrix} 6 9 3 -6 end{pmatrix} - begin{pmatrix} -6 -9 -3 6 end{pmatrix} begin{pmatrix} 0 0 0 0 end{pmatrix}

Since we found scalars and that satisfy the equation with at least one of them non-zero, the vectors and are linearly dependent.

Direct Observation of Linear Dependence

Additionally, we can observe that is a scalar multiple of :

mathbf{v_2} -3mathbf{v_1}

This directly confirms their linear dependence. Therefore, the vectors are indeed linearly dependent.

Conclusion

In conclusion, through both solving the system of equations and a direct observation, we have demonstrated that the vectors and are linearly dependent. Understanding these concepts and the methods to determine linear dependence is crucial for many areas of mathematics and its applications.