Kernel of Linear Transformation is Linear Subspace

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Theorem

Let $V, W$ be normed vector spaces on a field $F$.

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Let $T: V \to W$ be a linear transformation.

Then the kernel of $T$ is a linear subspace of $V$.

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Proof

By Kernel of Linear Transformation contains Zero Vector, $0_V \in \ker T$.

Closure under Addition

For $x, y \in \ker T$:

$\map T {x + y} = \map T x + \map T y = 0_W + 0_W = 0_W$

so:

$x + y \in \ker T$

so $\ker T$ is closed under addition.

$\Box$

Closure under Scalar Multiplication

For $k \in F, x \in \ker T$:

$\map T {k x} = k \map T x = k 0_W = 0_W$

so:

$k x \in \ker T$

So $\ker T$ is closed under scalar multiplication.

Hence the result by definition of linear subspace.

$\blacksquare$