Are Vectors Linearly Independent Calculator

Linear Independence Check:

Vectors v₁,...,v_k are linearly independent if det(G) ≠ 0 where G is Gram matrix, or rank of matrix with columns v_i is k.

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1. What is Linear Independence?

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. If vectors are dependent, at least one vector is redundant in the set.

2. How Does the Calculator Work?

The calculator checks linear independence by:

Creating a matrix with the vectors as columns
Performing Gaussian elimination to determine the matrix rank
Comparing the rank to the number of vectors

Key Concept: If rank = number of vectors, they are independent. Otherwise, they are dependent.

3. Importance of Linear Independence

Details: Linear independence is fundamental in linear algebra, affecting solutions to systems of equations, basis formation, and determining the dimension of vector spaces.

4. Using the Calculator

Tips: Enter vectors as comma-separated values (e.g., "1,2,3"). All vectors must have the same dimension. The calculator handles 2-4 vectors.

5. Frequently Asked Questions (FAQ)

Q1: What's the maximum number of independent vectors in Rⁿ?
A: The maximum is n. Any set with more than n vectors in Rⁿ must be dependent.

Q2: How is this different from orthogonal?
A: Orthogonal vectors are always independent, but independent vectors aren't necessarily orthogonal.

Q3: Can 2 vectors in R³ be independent?
A: Yes, two non-parallel vectors in R³ are independent, though they don't span the whole space.

Q4: What if one vector is all zeros?
A: Any set containing the zero vector is automatically dependent.

Q5: How does this relate to determinants?
A: For square matrices, non-zero determinant means the columns are independent. Our calculator works for non-square matrices too.