Exercise I.E.1

Question

Let $V$ be a vector space over a field $\mathbb{K}$, and let $\phi: V 	o V$ be a linear homomorphism. Show that for any $\lambda \in \mathbb{K}$ the eigenspace $V(\lambda)$ is a vector subspace of $V$.

Toğrul Topal · Accepted Answer

Let $v,v'\in V(\lambda)$ be arbitrary, and let $\alpha \in \mathbb{K}$. We then have
$$\phi(\alpha v + v') = \alpha\phi(v) + \phi(v') = \alpha\lambda v + \lambda v' = \lambda(\alpha v + v')$$
which proves that $\alpha v + v'$ belongs to the eigenspace of $\lambda$.

Exercise I.E.1

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Exercise I.E.1

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