Question
Explain in detail why the columns of $e^{t A}$ form a basis for the solution space to the system $\dot{\mathbf{u}}=A \mathbf{u}$.
Step 1
The matrix exponential $e^{tA}$ is defined as $e^{tA} = \sum_{n=0}^\infty \frac{(tA)^n}{n!}$. This series converges for all $t$ and for any square matrix $A$. The matrix $e^{tA}$ is fundamental in solving systems of linear differential equations. Show more…
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