1. What is Image Basis?

The image (or range) of a matrix A is the set of all possible linear combinations of its column vectors. A basis for the image consists of the linearly independent columns of A, which can be found by identifying the pivot columns after row reduction.

2. How Does the Calculator Work?

The calculator performs the following steps:

Where:

• Perform Gaussian elimination on the matrix
• Identify pivot columns in the row-reduced form
• Extract corresponding columns from the original matrix
• These columns form a basis for the image space

Explanation: The pivot columns after row reduction correspond to the linearly independent columns in the original matrix that span the image space.

3. Importance of Image Basis

Details: The image basis helps determine the rank of the matrix, understand the linear transformation represented by the matrix, and solve systems of linear equations.

4. Using the Calculator

Tips: Enter the matrix with rows separated by commas and elements separated by spaces. For example, "1 2 3, 4 5 6" represents a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between image and column space?
A: They are the same concept - the vector space spanned by the columns of the matrix.

Q2: How is image basis related to rank?
A: The number of vectors in the image basis equals the rank of the matrix.

Q3: Can the image basis be empty?
A: Only for the zero matrix, which has a trivial image space {0}.

Q4: Does the basis depend on the row reduction method?
A: The number of basis vectors is fixed, but the specific basis may vary depending on the reduction process.

Q5: How does this relate to solving Ax=b?
A: The system has a solution if and only if b is in the image of A.