Chapter 9.5, 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 in fact from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.

(a) If $A$ is an $n\times n$ matrix, then the vector differential equation $x^{\prime }=Ax$ cannot have $n$ linearly independent solutions.

(b) A cycle of generalized eigenvectors of $A$ corresponding to an eigenvalue $\lambda$ consisting of $k$ vectors yields $k$ linearly independent solutions to the vector differential equation $x^{\prime }=Ax$.

(c) The number of linearly independent solutions to $x^{\prime }=Ax$ corresponding to $\lambda$ is the eigenspace $E_{\lambda }$.

(d) If $v_0$ is an eigenvector of $A$ corresponding to $\lambda$ and $v_1$ is a vector satisfying $\left(A-\lambda I\right)v_1=v_0$, then $x\left(t\right)=e^{\lambda t}{v}_1$ is a solution to the vector differential equation $x^{\prime }=Ax$.