The Kernel of a Linear Transformation | Generated by AI
The kernel of a linear transformation is the set of all input vectors that the transformation maps to the zero vector.
Definition:
If ( T: V \to W ) is a linear transformation between two vector spaces ( V ) and ( W ), then the kernel (or null space) of ( T ) is:
[ \ker(T) = { v \in V \mid T(v) = 0 } ]
Key Properties:
- Subspace: The kernel of a linear transformation is always a subspace of the domain ( V ).
- Dimension: The dimension of the kernel is called the nullity of the transformation.
- Relation to Injectivity:
- If ( \ker(T) = {0} ), then ( T ) is injective (one-to-one).
- If ( \ker(T) ) has dimension greater than zero, then ( T ) is not injective.
Example:
Consider the matrix transformation ( T: \mathbb{R}^3 \to \mathbb{R}^2 ) given by:
[ T(x, y, z) = (x + y, y + z) ]
Writing this as a matrix,
[ T \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} ]
To find the kernel, solve:
[ \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} ]
This gives the system:
[ x + y = 0 ] [ y + z = 0 ]
Solving for ( x, y, z ):
[ x = -y, \quad z = -y ]
Thus, the kernel consists of all vectors of the form:
[ \ker(T) = \text{span} \left{ \begin{bmatrix} -1 \ 1 \ -1 \end{bmatrix} \right} ]
which is a one-dimensional subspace of ( \mathbb{R}^3 ).
Let me know if you need further clarification!