1. What is Base of Image?

The base of the image of a linear transformation T, denoted as base(im(T)), is the dimension of the image space, which equals the rank of the transformation matrix. It represents the number of linearly independent column vectors in the matrix.

2. How Does the Calculator Work?

The calculator computes the rank of the transformation matrix using Gaussian elimination:

Where:

• \( T \) — Linear transformation matrix
• \( \text{rank}(T) \) — Number of linearly independent columns/rows

Explanation: The calculator converts the matrix to row echelon form and counts the number of non-zero rows, which equals the rank.

3. Importance of Base Calculation

Details: Calculating the base of the image is fundamental in linear algebra for understanding the properties of linear transformations, solving systems of equations, and analyzing vector spaces.

4. Using the Calculator

Tips: Enter the transformation matrix with rows separated by semicolons and elements within rows separated by commas. For example: "1,2,3; 4,5,6" for a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between rank and nullity?
A: According to the rank-nullity theorem: rank(T) + nullity(T) = number of columns in T.

Q2: Can the rank exceed the matrix dimensions?
A: No, the rank cannot exceed the smaller of the number of rows or columns in the matrix.

Q3: What does a rank of zero indicate?
A: A rank of zero means the transformation maps all vectors to the zero vector (the zero transformation).

Q4: How is rank related to invertibility?
A: A square matrix is invertible if and only if its rank equals its dimension (full rank).

Q5: Does the rank depend on the field?
A: Yes, the rank of a matrix may vary depending on the field over which it's considered (e.g., real vs. complex numbers).