1. What is Basis and Dimension?

The dimension of a matrix space is its rank (number of linearly independent columns). A basis consists of the pivot columns from the original matrix after row reduction.

2. How Does the Calculator Work?

The calculator performs matrix operations to determine:

Where:

• \( A \) — Input matrix
• rank — Number of linearly independent columns
• pivot columns — Columns containing leading 1's in row echelon form

Explanation: The calculator performs Gaussian elimination to identify pivot columns and determine the matrix rank.

3. Importance of Basis and Dimension

Details: Understanding basis and dimension is crucial for solving linear systems, understanding vector spaces, and applications in computer graphics, machine learning, and physics.

4. Using the Calculator

Tips: Enter matrix values separated by commas for columns and semicolons for rows. All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dimension and rank?
A: For matrix spaces, dimension equals rank - both represent the number of linearly independent vectors.

Q2: How is the basis determined?
A: The basis consists of the original matrix columns corresponding to pivot positions after row reduction.

Q3: What's the maximum possible dimension?
A: The dimension cannot exceed the smaller of the row or column count (min(m,n)).

Q4: Does the calculator work for non-square matrices?
A: Yes, the calculator handles both square and rectangular matrices.

Q5: What about complex numbers?
This calculator currently only handles real numbers.