Matrix and eigenvalues of a given linear transformation

#linear_algebra

Suppose ${v_{1}, v_{2}, v_{3}}$ is a basis for $R^{3}$ and $T : R^{3} \to R^{3}$ is a linear transformation satisfying the following:

\begin{aligned} T (v_{1}) & = 4 v_{1} + 2 v_{2} \\ T (v_{2}) & = 5 v_{2} \\ T (v_{3}) & = - 2 v_{1} + 4 v_{2} + 5 v_{3} . \end{aligned}

Determine the eigenvalues of $T$ and find a basis for each eigenspace.

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Suppose $\{{\bf v}_1,{\bf v}_2,{\bf v}_3\}$ is a basis for ${\bf R}^3$ and $T:{\bf R}^3\to {\bf R}^3$ is a linear transformation satisfying the following:
\begin{align*}
	T({\bf v}_1) &= 4{\bf v}_1+2{\bf v}_2\\
	T({\bf v}_2) &= 5{\bf v}_2\\
	T({\bf v}_3) &= -2{\bf v}_1+4{\bf v}_2+5{\bf v}_3.
\end{align*}
Determine the eigenvalues of $T$ and find a basis for each eigenspace.