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,
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:
which is a one-dimensional subspace of \( \mathbb{R}^3 \).
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