1. What is a Basis of a Matrix?

The basis of a matrix refers to the set of linearly independent column vectors that span the column space of the matrix. These vectors form the fundamental building blocks of the vector space represented by the matrix.

2. How Does the Calculator Work?

The calculator uses Gaussian elimination to identify pivot columns:

Process:

1. Perform Gaussian elimination on the matrix
2. Identify columns containing pivots (leading 1's in row echelon form)
3. These columns form the basis for the column space

3. Importance of Basis Calculation

Details: Finding the basis is essential for understanding the fundamental structure of a matrix, solving systems of linear equations, and performing dimension analysis of vector spaces.

4. Using the Calculator

Tips: Enter your matrix with comma-separated values for each row, and semicolons between rows. For example: "1,2,3; 4,5,6" represents a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What exactly are pivot columns?
A: Pivot columns are those that contain the leading non-zero entry in each row after Gaussian elimination.

Q2: How many basis vectors will a matrix have?
A: The number of basis vectors equals the rank of the matrix.

Q3: Can a matrix have multiple different bases?
A: Yes, but all bases for a given space will have the same number of vectors (the dimension).

Q4: What's the difference between column space and row space basis?
A: This calculator finds the column space basis. For row space, you'd transpose the matrix first.

Q5: How is this related to linear independence?
A: The basis vectors are by definition linearly independent and span the space.