= Solution
{c}
Define the <mass matrix> $M_{ij}=\int_0^1\phi_j\phi_i$ and <stiffness matrix> $K_{ij}=\int_0^1\phi_j'\phi_i'$ for $1\leq i,j\leq N$. With coefficient vectors $U^m$, the step system is
$$
\boxed{\mathsf A U^{m+1}=M U^m,\qquad\mathsf A=M+\Delta t(K+M).}
$$
The matrices are symmetric, and $V^T\mathsf AV=\|v_h\|_2^2+\Delta t(\|v_h'\|_2^2+\|v_h\|_2^2)>0$ for nonzero $V$, proving positive definiteness.
For $h=1/2$, the first <basis> function is the full tent at $1/2$, and the last is the half tent ending at $1$. Integration gives
$$
M=\begin{pmatrix}1/3&1/12\\1/12&1/6\end{pmatrix},\qquad
K=\begin{pmatrix}4&-2\\-2&2\end{pmatrix}.
$$
Since $\Delta t=2/2=1$, the requested <matrix> in this normalization is
$$
\boxed{\mathsf A=2M+K=\begin{pmatrix}14/3&-11/6\\-11/6&7/3\end{pmatrix}.}
$$
Here the <matrix> symbol $M$ is distinct from the printed integer number of time steps.
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