1. What is Linear Independence?

A set of vectors (matrix columns) is linearly independent if no vector in the set can be written as a linear combination of the others. For a matrix, the columns are linearly independent if the rank of the matrix equals the number of columns.

2. How the Calculator Works

The calculator uses Gaussian elimination to determine the rank of the matrix and compares it to the number of columns:

Explanation: The rank represents the number of linearly independent rows or columns in the matrix. If this equals the total number of columns, they must all be independent.

3. Importance of Linear Independence

Details: Linear independence is fundamental in linear algebra, affecting solutions to systems of equations, basis determination, and matrix invertibility.

4. Using the Calculator

Steps:

1. Enter matrix dimensions (rows × columns)
2. Input all matrix elements
3. Click "Check Independence"
4. View results showing rank and independence status

5. Frequently Asked Questions (FAQ)

Q1: What does it mean if columns are dependent?
A: At least one column can be expressed as a combination of others, indicating redundancy in the matrix.

Q2: Can a matrix have more columns than rows and be independent?
A: No, the maximum rank is the smaller of rows or columns, so with more columns than rows, they must be dependent.

Q3: How is this related to matrix invertibility?
A: A square matrix is invertible if and only if its columns (and rows) are linearly independent.

Q4: What about row independence?
A: For any matrix, row rank equals column rank, so the same calculation applies to rows by transposing the matrix.

Q5: Does this work for complex matrices?
A: This calculator handles real numbers only, but the concept extends to complex numbers with appropriate modifications.