How to Prove Linear Independence?
To prove linear independence, set up the equation c1v1 + c2v2 + . . . + cnvn = 0. Organize the vectors into a matrix, solve the system AC=0, and determine if the only solution is c1 = c2 = . . . = cn = 0. If so, the vectors are linearly independent; otherwise, they are dependent.
Problem: Determine if the vectors are linearly independent.
- v1 : <1, 2>
- v2: <3, 4>
- v3: < 2, 5>
Solution:
We can construct a matrix with these vectors as columns and perform row reduction:
[Tex]\begin{bmatrix} 1 & 3 & 2\\ 2 & 4 & 5 \\ \end{bmatrix} [/Tex]
Performing row reduction, we find:
[Tex]\begin{bmatrix} 1 & 3 & 2\\ 0 & -2 & 1 \\\end{bmatrix} [/Tex]
Since the reduced matrix has a non-zero row, the vectors are linearly independent.
Linear Independence
Linear independence is a fundamental concept in mathematics that has numerous applications in fields like physics, engineering, and computer science. It is necessary for determining the size of a vector space and finding solutions for optimization problems.
In this article, we will learn about linear independence, providing a simple explanation of its applications. We will understand the necessary steps for testing linear independence, their significance in the context of vector spaces and matrices as well.
Table of Content
- What is Linear Independence?
- Steps to Determine Linear Independence
- Linear Independence in Vector Spaces
- Application of Linear Independence
- How to Prove Linear Independence?
- Conclusion: Linear Independence