Chapter 9.3, Problem 1TFR

True-False Review

For Questions (a)-(d), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem from the text. If true, provide an example, illustration, or brief explanation of why the statement is false.

(a) The set of all solutions to the vector differential equation $x^{\prime }\left(t\right)=A\left(t\right)x\left(t\right)+b\left(t\right)$ (where $A\left(t\right)$ is an $n\times n$ matrix function and $b\left(t\right)\ne 0$) is a vector space of dimension $n$.

(b) Any fundamental matrix for the vector differential equation $x^{\prime }=Ax$ is invertible.

(c) A fundamental solution set to a vector differential equation $x^{\prime }=Ax$ forms a spanning set for the vector space of all solutions to $x^{\prime }=Ax$.

(d) If the general solution to the homogeneous vector differential equation $x^{\prime }\left(t\right)=A\left(t\right)x\left(t\right)$ is $x_c\left(t\right)$ and $x_p\left(t\right)$ is a particular solution to the nonhomogeneous vector differential equation $x^{\prime }\left(t\right)=A\left(t\right)x\left(t\right)+x_p\left(t\right)$, then the general solution to the nonhomogeneous vector differential equation is $x_c\left(t\right)+x_p\left(t\right)$.