Basis of Image Calculator

Enter Matrix Elements

Matrix (comma separated rows):



Formula

To calculate the basis of an image in a linear transformation:

\[ B = \{v \in V : T(v) \neq 0\} \]

Where:

What is the Basis of Image?

A basis of an image, in the context of linear algebra, refers to a set of vectors that spans the image of a linear transformation or a matrix. These vectors are linearly independent, meaning they cannot be expressed as a linear combination of each other. The basis of an image provides a way to describe every vector in the image space in terms of a linear combination of the basis vectors.

Example Calculation

Let's assume the following matrix:

1, 2, 3
4, 5, 6
7, 8, 9

The basis of the image would be calculated by transforming this matrix into its row echelon form and extracting the non-zero rows.

Using the row echelon form, the basis vectors are:

1, 2, 3
0, 1, 2

The basis of the image is \(\{(1, 2, 3), (0, 1, 2)\}\).