Calculate a basis for the image of a linear transformation from its matrix, identifying pivot column vectors and the rank or dimension of Im(T).

Basis Of Image Calculator

Vector Space & Transformation
Matrix Input

Enter the transformation matrix (with respect to the standard bases, or the bases you are using) to compute a basis for the image Im(T) (the column space).

Basis Of Image Formula

The image (also called the range) of a linear transformation T is the set of all outputs T(v). A basis of the image is any linearly independent set of vectors that spans this image.

Im(T) = \ T(v): v V \ if A is the matrix of T, Im(T) = Col(A)

Variables:

  • Im(T) is the image (range) of the transformation T
  • V is the domain vector space
  • T(v) is the output of the linear transformation applied to the input vector v
  • A is a matrix representation of T (with respect to chosen bases); the image is the column space Col(A)

To find a basis of the image, apply T to a spanning set (typically a basis) of V to get vectors in the codomain, then keep a linearly independent subset that still spans the same set of outputs. If T is given by a matrix A, a standard method is to row-reduce A to find the pivot columns; the corresponding columns of the original matrix form a basis for the image (the column space).

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 a common way to calculate a basis of the image.

  1. Identify the linear transformation T:V\to W (or its matrix A with respect to chosen bases).
  2. Choose a basis {v1,…,vn} of the domain V.
  3. Compute the images T(v1),…,T(vn), which are vectors in the codomain W.
  4. Form a matrix whose columns are these vectors (equivalently, if you already have the matrix A, its columns are T(e1),…,T(en) in standard coordinates).
  5. Row-reduce to find which columns are pivot columns, then take the corresponding columns of the original matrix as a basis for Im(T). The number of basis vectors is the rank (dimension of the image).

Example Problem:

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

Let {v1, v2, v3, v4} be a basis of a vector space V.

Define the linear transformation T(v)=2v (over the real numbers, or any field where 2 ≠ 0).

Then Im(T)=V, and one basis of the image is {T(v1),T(v2),T(v3),T(v4)} = {2v1,2v2,2v3,2v4} (which is also a basis of V).