To calculate the basis of an image in a linear transformation:
\[ B = \{v \in V : T(v) \neq 0\} \]
Where:
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.
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 \(\{(1, 2, 3), (0, 1, 2)\}\).