I am reading the book on Introduction to Differential Geometry by Boothby. I am unable to understand a particular concept related to Differentiation of vector fields ((Pg. 310) Chapter VII Eq 2.10)
Let $Y$ be tangent vector field on a Manifold $M$ of dimension $m$ in $\mathbb{R}^n$. Let the local coordinates be $u=(u_1,\dots,u_m)$. Let $F_1,\dots, F_m$ be the vectors that span the tangent space of $M$. We can denote by $Y$ as follows: \begin{align} Y = \sum_{i} b^i(u) F_i \end{align}
Let $X_{p}$ be a tangent vector at $p$, $X_{p}=\sum a^{j} F_{j p}$
Now choose any differentiable curve $p(t)$ with $p\left(t_{0}\right)=p$ and $(d p / d t)_{t_{0}}=X_{p}$, so that in local coordinates it is defined by $u(t)=\left(u^{1}(t), \ldots, u^{m}(t)\right)$ with $ u^{i}\left(t_{0}\right)=u_{0}^{i} \text { and }\left(d u^{i} / d t\right)_{t_{0}}=a^{i}$
Then the books says we can write \begin{align} \left(\frac{d b^{k}}{d t}\right)_{t_{0}}=\sum_{j=1}^{m}\left(\frac{\partial b^{k}}{\partial u^{j}}\right)_{u_{0}} a^{j}=X_{p} b^{k} \end{align}
I cannot understand how the second equality follows. Moreover, $\left(\frac{d b^{k}}{d t}\right)_{t_{0}}$ is a scalar and $X_p b^k$ is a vector. How can they be equal?