1. What is Row Space?

The row space of a matrix is the set of all possible linear combinations of its row vectors. It represents the subspace spanned by the rows of the matrix in the vector space.

2. How Does the Calculator Work?

The calculator performs the following steps:

1. Converts the input matrix to row echelon form (REF) using Gaussian elimination
2. Identifies all non-zero rows in the REF matrix
3. Returns these non-zero rows as the basis for the row space

3. Importance of Row Space

Details: The row space is fundamental in linear algebra. It helps determine:

• The rank of a matrix (dimension of row space)
• Linear independence of vectors
• Solutions to systems of linear equations

4. Using the Calculator

Tips: Enter your matrix with rows separated by semicolons and columns separated by commas. For example:
1,2,3;4,5,6;7,8,9

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between row space and column space?
A: Row space is spanned by row vectors, column space by column vectors. They have the same dimension (matrix rank).

Q2: How is row space related to rank?
A: The dimension of the row space equals the rank of the matrix.

Q3: Can row space be empty?
A: No, even a zero matrix has a row space (the zero vector space).

Q4: Does row space change with elementary row operations?
A: No, elementary row operations preserve the row space.

Q5: How is row space used in solving linear systems?
A: The row space helps determine if a solution exists and the nature of solutions.