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 image
- V is the vector space
- T(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}
- First, identify the vector space V.
- Next, determine the linear transformation T(v).
- Next, apply the transformation T(v) to each vector v in V.
- Identify the vectors v in V for which T(v) ≠ 0.
- 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
