1. What is Basis of Column Space?

The basis for the column space of a matrix A consists of the pivot columns of A. These columns form a linearly independent set that spans the column space of A, which is the vector space formed by all linear combinations of A's columns.

2. How Does the Calculator Work?

The calculator performs matrix reduction to identify pivot columns:

Steps:

1. Convert matrix to row echelon form
2. Identify columns with leading 1's (pivot columns)
3. Extract corresponding columns from original matrix

3. Importance of Column Space Basis

Applications: Finding a basis for the column space is fundamental in linear algebra for determining rank, solving linear systems, and understanding linear transformations.

4. Using the Calculator

Instructions: Enter your matrix with each row on a new line and elements separated by spaces. The calculator will return the basis vectors for the column space.

5. Frequently Asked Questions (FAQ)

Q1: How is this different from row space basis?
A: Row space basis comes from non-zero rows of row echelon form, while column space basis comes from pivot columns of the original matrix.

Q2: What if my matrix has no pivot columns?
A: Then the column space is just the zero vector space, and has an empty basis.

Q3: Does the order of basis vectors matter?
A: The order isn't unique, but the number of basis vectors (the rank) is always the same.

Q4: Can I use this for complex matrices?
A: This calculator handles real matrices. Complex matrices would require complex arithmetic.

Q5: How accurate are the results?
A: Results are subject to floating-point precision limitations in PHP.