Chapter 9.8, Problem 1TFR

To determine

(a)

To find:

Whether the derivative of the matrix exponential function $e^{At}$ with respect to the variable $t$ is $A{e}^{At}$. is true or false.

To determine

(b)

To find:

Whether the transition matrix $X_0\left(t\right)$ for the vector differential equation $x^{\prime }=Ax$ is precisely the same as the matrix exponential function $e^{At}$ is true or false

To determine

(c)

To find:

Whether if the matrix exponential function $e^{At}$ is known, then one can explicitly solve the problem $x^{\prime }=Ax$, $x(0)=x_0$ is true or false.

To determine

(d)

To find:

Whether the transition matrix for a linear system is always invertible is true or false.

To determine

(e)

To find:

Whether the matrix exponential function $e^{At}$ can be written as $e^{At}=X\left(t\right)X^{-1}(0)$ for any fundamental matrix $X\left(t\right)$ for the vector differential equation $x^{\prime }=Ax$ is true or false.

To determine

(f)

To find:

Whether if $\left\{v_1,v_2,\cdots ,v_n\right\}$ is linearly independent in ${ℝ}^n$, then so is $\left\{e^{At}{v}_1,e^{At}{v}_2,\cdots ,e^{At}{v}_n\right\}$ for all $n\times n$ matrices $A$ is true or false.