1. What is Linear Dependence?

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

2. How the Calculator Works

The calculator checks for linear dependence by:

Steps:

1. Creates a matrix with input vectors as columns
2. Computes the matrix rank using Gaussian elimination
3. Compares rank to number of vectors

3. Importance of Linear Dependence

Applications: Determining linear dependence is fundamental in linear algebra, used for solving systems of equations, analyzing vector spaces, and more.

4. Using the Calculator

Instructions: Enter vectors separated by semicolons, with components separated by commas. Example: "1,2,3; 4,5,6; 7,8,9" for three 3D vectors.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between linear dependence and independence?
A: Dependent vectors have redundancy (at least one is a combination of others), while independent vectors provide unique information.

Q2: Can two vectors be linearly dependent?
A: Yes, if they are scalar multiples of each other (collinear).

Q3: How many vectors in R^n can be independent?
A: At most n vectors can be linearly independent in R^n.

Q4: What does zero determinant indicate?
A: For square matrices, zero determinant means the columns (or rows) are linearly dependent.

Q5: Can linearly dependent vectors span a space?
A: They can span a space, but not efficiently - some vectors are redundant.