1. What is the Image of a Matrix?

The image (or range) of a matrix A is the set of all possible outputs of the linear transformation represented by A. It's the span of the column vectors of A. The image basis consists of linearly independent columns from A that correspond to pivot columns in the reduced row echelon form (RREF) of A.

2. How Does the Calculator Work?

The calculator performs these steps:

Mathematically: \[ \text{Image}(A) = \{ A\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^n \} = \text{Span}\{\text{Columns of } A\} \]

3. Importance of Image Basis

Details: Finding the image basis helps determine the output space of a linear transformation, understand the transformation's rank, and analyze solutions to linear systems.

4. Using the Calculator

Tips: Enter your matrix with rows separated by semicolons and elements separated by spaces or commas. For example:

1 2 3; 4 5 6; 7 8 9

represents a 3×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between image and kernel?
A: The image is the output space (range) of the transformation, while the kernel is the set of inputs that map to zero (null space).

Q2: How does this relate to matrix rank?
A: The dimension of the image space equals the rank of the matrix.

Q3: What if my matrix has complex numbers?
A: This calculator currently only handles real-valued matrices.

Q4: Why use original columns instead of RREF columns?
A: The RREF columns may not be in the original space - we need the original columns that correspond to the pivot positions.

Q5: Can I use this for non-square matrices?
A: Yes, the calculator works for any m×n matrix.