1. What is Basis for 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 (range) of the matrix.

2. How Does the Calculator Work?

The calculator performs the following steps:

Key Concept: The pivot columns of A form a basis for Col A, while the pivot columns of Aᵀ form a basis for Row A.

3. Importance of Column Space Basis

Applications: Finding a basis for the column space is essential for determining the rank of a matrix, solving linear systems, and understanding the range of a linear transformation.

4. Using the Calculator

Instructions: Enter matrix elements separated by spaces, with rows separated by semicolons. For example:
1 2 3; 4 5 6; 7 8 9

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between column space and row space?
A: Column space is the span of the columns, while row space is the span of the rows. They have the same dimension (rank).

Q2: Can the basis vectors be different for the same column space?
A: Yes, there are infinitely many bases for any given space, but they must all have the same number of vectors (the rank).

Q3: How does this relate to solving Ax=b?
A: The system has a solution if and only if b is in the column space of A.

Q4: What about matrices with complex numbers?
A: The same principles apply, though this calculator currently handles real numbers only.

Q5: How is this different from null space?
A: Column space contains all possible outputs (Ax), while null space contains all solutions to Ax=0.