Enter the vector space and linear transformation into the calculator to determine the basis of the image.

## Basis Of Image Formula

The following formula is used to calculate the basis of an image in a linear transformation.

B = {v ∈ V : T(v) ≠ 0}

Variables:

• B is the basis of the imageV is the vector spaceT(v) is the linear transformation of vector v

To calculate the basis of an image, you need to find all vectors in the vector space V that, when transformed by T, do not result in the zero vector. These vectors form the basis of the image under the transformation T.

## 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.

## How to Calculate Basis Of Image?

The following steps outline how to calculate the Basis Of Image using the formula: B = {v ∈ V : T(v) ≠ 0}

1. First, identify the vector space V.
2. Next, determine the linear transformation T(v).
3. Next, apply the transformation T(v) to each vector v in V.
4. Identify the vectors v in V for which T(v) ≠ 0.
5. Finally, compile the vectors v that satisfy the condition T(v) ≠ 0 to form the basis of the image B.

Example Problem:

Use the following variables as an example problem to test your knowledge:

Vector space V = {v1, v2, v3, v4}

Linear transformation T(v) = 2v