1. What is Column Space?

The column space (or range) of a matrix is the set of all possible linear combinations of its column vectors. It represents all vectors that can be obtained by multiplying the matrix with some vector.

2. How Does the Calculator Work?

The calculator uses row reduction (Gaussian elimination) to:

Key Concept: The pivot columns in the row echelon form indicate which original columns are linearly independent and form a basis for the column space.

3. Importance of Column Space

Applications: Column space is fundamental in linear algebra for solving linear systems, understanding matrix transformations, and analyzing vector spaces.

4. Using the Calculator

Instructions: Enter matrix elements separated by commas for columns and semicolons for rows. Example: "1,2,3;4,5,6" creates a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

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

Q2: How is column space related to solutions of Ax=b?
A: The system Ax=b has a solution if and only if b is in the column space of A.

Q3: What does zero column space mean?
A: A zero column space (only the zero vector) means the matrix is the zero matrix or all columns are linearly dependent.

Q4: Can column space be larger than the number of rows?
A: No, the dimension of column space (rank) cannot exceed the number of rows or columns.

Q5: How is column space used in real applications?
A: It's used in data analysis, computer graphics, engineering systems, and anywhere linear transformations are modeled.