= Solution
Use the <Riemann curvature tensor> convention $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$. A <Jacobi field> is a smooth <vector field> $J$ along $\gamma$ satisfying
$$
\boxed{D_t^2J+R(J,\dot\gamma)\dot\gamma=0,}
$$
where $D_t$ is the <covariant derivative along a curve> for the <Levi-Civita connection>. Differentiating a <geodesic variation> gives this equation, and conversely every <Jacobi field> arises from a <geodesic variation> by varying its initial point and velocity.
Choose a <parallel frame> $E_1(t),\ldots,E_n(t)$ along $\gamma$ and write $J=\sum_a y_aE_a$. The <Jacobi equation> becomes the linear system
$$
y''(t)+K(t)y(t)=0,\qquad K_{ab}(t)=\langle R(E_b,\dot\gamma)\dot\gamma,E_a\rangle.
$$
The existence and uniqueness theorem for linear <ordinary differential equations> says that each pair $(y(0),y'(0))\in\mathbb R^n\oplus\mathbb R^n$ determines exactly one solution on $[0,1]$. Addition and scalar multiplication preserve the equation. Thus evaluation of initial position and covariant velocity is a linear isomorphism
$$
\mathcal J_\gamma\longrightarrow T_{\gamma(0)}M\oplus T_{\gamma(0)}M,
\qquad J\longmapsto(J(0),D_tJ(0)),
$$
and \b[the dimension is]
$$
\boxed{\dim\mathcal J_\gamma=2n.}
$$
For the convexity assertion, use the convex-normal meaning of a geodesically convex open set: its points are joined by a unique geodesic within the set, and the joining geodesic depends smoothly on its endpoints. Equivalently, the appropriate star-shaped restriction of $\exp_p$ is a <diffeomorphism> onto the set. The convex-normal-neighbourhood theorem supplies such sets around every point. Mere existence of some minimizing geodesic, without this uniqueness and normality, would not imply the assertion.
Let $p=\gamma(0)$ and $v=\dot\gamma(0)$. For a <Jacobi field> with $J(0)=0$ and $D_tJ(0)=w$, differentiating $\exp_p(t(v+sw))$ gives the standard differential-of-the-<exponential map> identity
$$
J(1)=(d\exp_p)_v(w).
$$
In a <convex normal neighbourhood>, $(d\exp_p)_v$ is invertible. Therefore a <Jacobi field> vanishing at both endpoints has $w=0$ and hence is identically zero. The difference of two fields with the same endpoint values consequently vanishes. More precisely, the map
$$
\boxed{\mathcal J_\gamma\longrightarrow T_pM\oplus T_{\gamma(1)}M,\qquad J\longmapsto(J(0),J(1))}
$$
is an isomorphism: it is injective and both spaces have dimension $2n$. The constant-geodesic case follows directly from $D_t^2J=0$. This uses the convex-normal-neighbourhood theorem, the differential-of-the-<exponential map> identity, and linear <ordinary differential equation> uniqueness.
For the <special unitary group> example, use <Jacobi fields from conjugation at a central endpoint>. Write
$$
A=\operatorname{diag}(2\pi i/3,2\pi i/3,-4\pi i/3),\qquad z=e^{2\pi i/3}.
$$
The endpoints of $\gamma(t)=\exp(tA)$ are $I$ and $zI$. The latter is central. For every $X\in\mathfrak{su}(3)$ consider
$$
F_X(s,t)=e^{sX}e^{tA}e^{-sX}.
$$
Conjugation is an <isometry> for a <bi-invariant Riemannian metric>, and <geodesics of a bi-invariant metric are one-parameter subgroups>. Thus $F_X$ is a <geodesic variation>. Its <Jacobi field> is
$$
J_X(t)=Xe^{tA}-e^{tA}X,
$$
which vanishes at $t=0$ and $t=1$ because both endpoints are central. Its initial covariant derivative is $[X,A]$.
The linear map $X\mapsto J_X$ has kernel equal to the <centralizer of an element of a Lie algebra> of $A$. Indeed, $J_X\equiv0$ implies $[X,A]=0$ on differentiating at $t=0$, and that commutation conversely implies $J_X\equiv0$. The repeated first two eigenvalues give
$$
\mathfrak z(A)=\left\{\begin{pmatrix}B&0\\0&c\end{pmatrix}:B\in\mathfrak u(2),\ c\in i\mathbb R,\ \operatorname{tr}B+c=0\right\},\qquad\dim_\mathbb R\mathfrak z(A)=4.
$$
As $\dim_\mathbb R\mathfrak{su}(3)=8$, <rank-nullity theorem> gives
$$
\boxed{\dim\{J:J(0)=J(1)=0\}\geq8-4=4.}
$$
Explicit independent generators are the fields belonging to $E_{13}-E_{31}$, $i(E_{13}+E_{31})$, $E_{23}-E_{32}$ and $i(E_{23}+E_{32})$. Their initial derivatives $[X,A]$ are independent because the eigenvalue differences in the $13$ and $23$ entries are $2\pi i$. This also demonstrates directly why the endpoint-value conclusion fails along this geodesic.
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