1. What is Column Space?

The column space (col(A)) of a matrix is the vector space spanned by its column vectors. It represents all possible linear combinations of the matrix's columns and is fundamental in linear algebra.

2. How the Calculator Works

The calculator finds a basis for the column space by:

Steps:

• 1. Convert matrix to reduced row echelon form (RREF)
• 2. Identify pivot columns
• 3. Extract corresponding columns from original matrix

3. Importance of Column Space

Applications: Column space is essential for solving linear systems, understanding linear transformations, and in applications like machine learning and computer graphics.

4. Using the Calculator

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

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: Ax=b has a solution if and only if b is in the column space of A.

Q3: What does the dimension of column space represent?
A: The dimension is the rank of the matrix - the number of linearly independent columns.

Q4: Can column space be larger than the number of rows?
A: No, the dimension is at most the smaller of the row or column count.

Q5: How does column space relate to image of a linear transformation?
A: The column space is exactly the image of the linear transformation represented by the matrix.