Matrix Basis Calculator

Matrix Basis:

\[ \text{Basis} = \{\text{Pivot columns of } A\} \text{ for column space} \] \[ \text{Basis} = \{\text{Non-zero rows of } R\} \text{ for row space} \]

Basis for Selected Space:

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1. What is a Matrix Basis?

A basis is a set of linearly independent vectors that span a vector space. For matrix spaces, the basis can be found from the pivot columns (column space), non-zero rows of RREF (row space), or solutions to Ax=0 (null space).

2. How Does the Calculator Work?

The calculator performs matrix operations to find:

\[ \text{Column space basis} = \{\text{Pivot columns of original matrix } A\} \] \[ \text{Row space basis} = \{\text{Non-zero rows of RREF } R\} \] \[ \text{Null space basis} = \{\text{Special solutions to } Ax=0\} \]

Explanation: The calculator performs Gaussian elimination to find the reduced row echelon form (RREF) and extracts the basis vectors accordingly.

3. Importance of Basis Calculation

Details: Finding a basis helps determine the dimension of matrix spaces, understand linear transformations, and solve systems of linear equations.

4. Using the Calculator

Tips: Enter matrix elements separated by commas, rows separated by semicolons. For example: "1,2,3;4,5,6;7,8,9" for a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between column and row space?
A: Column space consists of all linear combinations of columns, while row space consists of all linear combinations of rows.

Q2: How is null space related to these?
A: Null space contains all solutions to Ax=0, and its dimension plus rank equals the number of columns.

Q3: Can the calculator handle complex numbers?
A: This version only handles real numbers. Complex matrices would require additional functionality.

Q4: What if my matrix is not full rank?
A: The calculator will correctly identify dependent vectors and only include independent ones in the basis.

Q5: How accurate are the calculations?
A: Results are limited by floating-point precision. For exact arithmetic, symbolic computation would be needed.