Linear Algebra Knowledge Evaluation

You are tasked with answering a series of linear algebra problems that assess your understanding of vector spaces, matrix operations, and system of equations. Answer each question with clear, detailed explanations.

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\section*{Linear Algebra Exam}

\begin{enumerate}

\item Answer the following \textbf{TRUE/FALSE} questions. You must write the entire word \textbf{TRUE} or \textbf{FALSE}.

\begin{enumerate}

\item The vectors $\{[1, 0, 1], [2, 0, 2], [0, 1, 0]\}$ are linearly independent.

\item The dimension of the span of $\{[1, 0, 1], [2, 0, 2], [0, 1, 0]\}$ is 3.

\item The functions $\{1, 1 + t, 2 + t\}$ are linearly independent.

\item If $Ax = 0$ has a solution $x \neq 0$ then $Ax = b$ must have infinitely many solutions for any $b$.

\item Four vectors in $\mathbb{R}^3$ are necessarily linearly dependent.

\item Four vectors in $\mathbb{R}^3$ necessarily span $\mathbb{R}^3$.

\item The set of solutions of the differential equation $y' + ty = t$ forms a vector space.

\end{enumerate}

\item Consider the linear system of equations

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