Linear Independence in Vector Spaces
Vectors are considered linearly independent if no vector in a set can be represented as a linear combination of the others. In other words, a set of vectors {v1, v2, . . . , vn} is linearly independent if the only solution to the equation:
c1v1 + c2v2 + . . . + cnvn = 0
(where c1, c2 . . . cn are scalars, not all zero) is the trivial solution where all scalars are zero.
Examples of Linear Independence in Vectors
Consider a set of vectors in ℝ³: {(1, 0, 0), (0, 1, 0), (0, 0, 1)}.
These vectors are linearly independent because no vector can be expressed as a linear combination of the others.
Now consider, {(1, 0, 0), (2, 0, 0), (3, 0, 0)}. It would be linearly dependent, as the third vector is a scalar multiple of the first.
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