Give the definition for a function \(L: \mathbb{R}^n \to \mathbb{R}^m\) to be a linear transformation.
Prove that the kernel of a linear transformation \(L: \mathbb{R}^n \to \mathbb{R}^m\) is a subspace of \(\mathbb{R}^n\).
Solution
Goal. Define a linear transformation and prove its kernel is a subspace.
1. (a) \(L: \RR^n \to \RR^m\) is linear iff \(L(x + y) = L(x) + L(y)\) and \(L(cx) = cL(x)\) for all \(x, y \in \RR^n\) and \(c \in \RR\).
definition of a linear transformation.
2. (b) \(\ker L = \theset{x \in \RR^n : L(x) = 0}\) is a subspace.
2.1. \(0 \in \ker L\).
\(L(0) = L(0 + 0) = L(0) + L(0)\), so \(L(0) = 0\).
2.2. Closed under addition: if \(x, y \in \ker L\), then \(L(x + y) = L(x) + L(y) = 0 + 0 = 0\), so \(x + y \in \ker L\).
linearity.
2.3. Closed under scalar multiplication: if \(x \in \ker L\) and \(c \in \RR\), then \(L(cx) = cL(x) = c \cdot 0 = 0\), so \(cx \in \ker L\).
linearity.
2.4. Hence \(\ker L\) is a subspace.
2.1, 2.2, 2.3 verify the three subspace axioms.
3. Q.E.D.
2.4 is the claim.