Understanding and Calculating Kernel, Range, Rank, and Nullity of a Linear Transformation

In this article, we will discuss and calculate the kernel, range, rank, and nullity of a given linear transformation. Specifically, we will analyze the transformation T: R^2 → R0 defined by T(x, y) xy. These concepts are fundamental in understanding the behavior and properties of linear transformations in linear algebra.

Introduction to Linear Transformations

A linear transformation T: V → W between two vector spaces V and W is a function that preserves the operations of vector addition and scalar multiplication. In simpler terms, for all vectors v, w ∈ V and scalars α, the transformation T satisfies:

T(v w) T(v) T(w) T(αv) αT(v)

In this article, we will focus on the linear transformation T: R^2 → R defined by T(x, y) xy, where x, y ∈ R.

Step 1: Kernel

The kernel of a linear transformation T is the set of all vectors v in the domain such that T(v) 0. For our transformation T: R^2 → R given by T(x, y) xy, we find the kernel as follows:

For the kernel:

T(x, y)  x * y  0

To solve for the kernel, we solve the equation:

x * y  0

This implies y -x. Thus, the kernel is given by:

Ker T  {(x, -x) | x ∈ R}

This can also be expressed as:

Ker T  span{(1, -1)}

The kernel is a one-dimensional subspace of R^2.

Step 2: Range

The range of a linear transformation T is the set of all possible outputs of the transformation. For our transformation T(x, y) xy, we find the range as follows:

T(x, y)  x * y

The output T(x, y) can take any real number value as x and y range over all real numbers. Therefore, the range is:

Range T  R

This is a one-dimensional space.

Step 3: Rank

The rank of a linear transformation is the dimension of its range. Since the range of T is R, which is one-dimensional, we have:

Rank T  1

Step 4: Nullity

The nullity of a linear transformation is the dimension of its kernel. The kernel of T is spanned by one vector, so:

Nullity T  1

Summary

Kernel:

Ker T  span{(1, -1)}Dimension: 1

Range:

Range T  RDimension: 1

Rank:

Rank T  1

Nullity:

Nullity T  1

These results are consistent with the Rank-Nullity Theorem which states that:

Rank T   Nullity T  dim(domain)  2

In this case:

1   1  2

This confirms our calculations are correct.

Additional Notes

Let f: V to W be a linear transformation. The kernel of f is defined as:

ker f  {x in V: fx  0}

The range or image of f is:

im f  {w in W: w  fx, x in V}

The rank of f is the dimension of the range, denoted as:

dim im f

The rank-nullity theorem states:

dim ker f   dim im f  dim V

For the given transformation T: R^2 → R, we found that:

ker T  left{left[  begin{matrix}  x  -x  end{matrix}  right] : x in R  right}im T  R

The rank is 1, and the rank-nullity is 1, which confirms the results from the calculations above.

Understanding these concepts is crucial for analyzing linear transformations and their properties. By applying these principles, you can gain insights into the nature of linear transformations and their significance in various mathematical contexts.