Understanding and Calculating Rank and Nullity of Linear Transformations

Linear transformations are fundamental in linear algebra with wide applications in various fields including computer science, physics, and engineering. One crucial aspect of understanding linear transformations is calculating their rank and nullity. This article will guide you through the process of finding the rank and nullity of a linear transformation, from definitions to example applications.

Definitions

Rank: The rank of a linear transformation T: V to W is the dimension of the image of T, which is the subspace of W that T maps into.

Nullity: The nullity of T is the dimension of the kernel of T, which is the subspace of V that maps to the zero vector in W.

Steps to Compute Rank and Nullity

1. Represent the Linear Transformation as a Matrix

If T is represented in a finite-dimensional vector space V with a basis, you can express T as a matrix A by choosing bases for V and W.

2. Find the Kernel

- Solve the equation Tv 0 or Amathbf{x} mathbf{0} if using a matrix. - The kernel consists of all vectors v in V such that Tv 0. - Find a basis for the kernel and count the number of vectors in this basis to determine the nullity.

3. Find the Image

- The image of T is the set of all vectors that can be expressed as Tv for v in V, or the column space of A. - To find a basis for the image, reduce the matrix A to its row echelon form or reduced row echelon form using Gaussian elimination. - The number of non-zero rows in the row echelon form corresponds to the dimension of the image, which gives the rank.

4. Apply the Rank-Nullity Theorem

The Rank-Nullity Theorem states that for a linear transformation T: V to W: dimV rankT nullityT

where dimV is the dimension of the domain V.

Example

Consider a linear transformation T: R^3 to R^2 given by the matrix:

A  begin{pmatrix} 1  2  3  0  1  4 end{pmatrix}

Find the Kernel

Solve Amathbf{x} mathbf{0}:

begin{pmatrix} 1  2  3  0  1  4 end{pmatrix} begin{pmatrix} x_1  x_2  x_3 end{pmatrix}  begin{pmatrix} 0  0 end{pmatrix}

This gives the system of equations:

x_1 2x_2 3x_3 0
x_2 4x_3 0

From this we can express x_2 and x_1 in terms of x_3. Let x_3 t:

x_2 -4t
x_1 5t

Hence the kernel is spanned by begin{pmatrix} 5 -4 1 end{pmatrix}, so the nullity is 1.

Find the Image

The row echelon form of A is:

begin{pmatrix} 1  2  3  0  1  4 end{pmatrix}

There are 2 non-zero rows so the rank is 2.

Conclusion

Using the Rank-Nullity Theorem:

dimR^3 3 rankT nullityT rarr; 3 2 1

Thus, the rank of the transformation is 2 and the nullity is 1.