# Image of a Linear Operators Eigenspace

Given a Linear Operator T and a vector or linear combination of vectors from within a single eigenspace of T, will those vectors always be projected back into that eigenspace when given as input to T? I tend to think yes since that kind of seems to be the definition of an eigenvector.

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Supposing that $\vec v_1,...,\vec v_n$ are vectors in the eigenspace of $T$ corresponding to the eigenvalue $\lambda$, and if $a_1,...,a_n$ are constants, then we have $$T(a_1\vec v_1+...+a_n\vec v_n)=a_1T(\vec v_1)+...+a_nT(\vec v_n)=a_1\lambda\vec v_1+...+a_n\lambda\vec v_n=\lambda(a_1\vec v_1+...+a_n\vec v_n),$$ so you're correct.
Yes. Let $v_i$ be an eigenvector to the eigenvalue $\lambda_i$. Let $x=\sum_{i=1}^n \mu_i v_i$, then $Tx = \sum_{i=1}^n \lambda_i \mu_i v_i$.
To prove your case, let $\lambda_i = \lambda_j =: \lambda$ for $i,j$. Then $Tx = \lambda x$ by the above calculation.