1. What is Matrix Image?

The image (or range) of a matrix A is the set of all possible outputs of the linear transformation represented by A. Formally, for an m×n matrix A, the image is defined as:

2. How Does the Calculator Work?

The calculator finds a basis for the image of the matrix by:

1. Performing Gaussian elimination to reduce the matrix to row-echelon form
2. Identifying the pivot columns
3. Extracting the corresponding columns from the original matrix
4. These columns form a basis for the image space

3. Importance of Matrix Image Calculation

Details: The image of a matrix represents the output space of the linear transformation. Understanding the image helps determine whether a system of equations has solutions, analyze the properties of linear transformations, and solve various problems in engineering and physics.

4. Using the Calculator

Tips: Enter the matrix with rows separated by newlines and elements within each row 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 image and kernel of a matrix?
A: The image is the set of all outputs (range), while the kernel is the set of all inputs that map to zero (null space).

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

Q3: Can the image be the entire space?
A: Yes, if the matrix has full row rank, then its image is the entire output space.

Q4: What if my matrix has complex numbers?
A: This calculator currently only supports real-valued matrices.

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