I'm reading the tutorial of MPM in physic based simulation (https://www.math.ucla.edu/~cffjiang/research/mpmcourse/mpmcourse.pdf). I encountered some mathematical problem when reading the section 5 Kinematics Theory, equation 32, 33. To be specific: $$J\mathrm{d}\mathbf{S}\cdot\mathrm{d}\mathbf{L}=\mathrm{d}\mathbf{s}\cdot(\mathbf{F}\mathrm{d}\mathbf{L})\tag{32}$$ Here $J=\det\mathbf{F}$ is the Jacobian determinate, and $\mathbf{F}$ is the $3\times3$ Jacobi matrix of a map $\mathbf{\Phi}$ from material space $\Omega^0\subset\mathbb{R}^3$ to world space $\Omega^t\subset\mathbb{R}^3$. $\mathrm{d}\mathbf{S}$ and $\mathrm{d}\mathbf{s}$ represent tiny areas with directions in $\Omega^0$ and $\Omega^t$, respectively, and $\mathrm{d}\mathbf{L}$ is a tiny vector in $\Omega^0$.
In the tutorial, it states that equation (32) need to be true for all $\forall\mathrm{d}\mathbf{L}\in\Omega^0$, therefore we can have the relationship between $\mathrm{d}\mathbf{s}$ and $\mathrm{d}\mathbf{S}$ in equation (33):$$\mathrm{d}\mathbf{s}=\mathbf{F}^{-\mathsf{T}}J\mathrm{d}\mathbf{S}\tag{33}$$
I don't see why this is true. I tried to substitute $\mathrm{d}\mathbf{s}$ term in (32) with (33), but I seem to see no difference.
Can somebody explain to me why (32) is true for all dL leads to (33)? Any help would be appreciated!
One notice that $$\mathbf{x}\cdot \mathbf{y}=\mathbf{y}^\mathsf{T}\mathbf{x}.$$ Thus from equation (32), we could show that:\begin{equation} \begin{aligned} J\mathrm{d}\mathbf{S}\cdot\mathrm{d}\mathbf{L}&=\mathrm{d}\mathbf{s}\cdot(\mathbf{F}\mathrm{d}\mathbf{L})\\ &=\mathrm{d}\mathbf{L}^\mathsf{T}\mathbf{F}^\mathsf{T}\mathrm{d}\mathbf{s}\\ &=\mathbf{F}^\mathsf{T}\mathrm{d}\mathbf{s}\cdot\mathrm{d}\mathbf{L} \end{aligned} \end{equation} With two matrices $A$ and $B$, if $A\cdot\mathbf{x}=B\cdot\mathbf{x}$ for all $\mathbf{x}$, then $A=B$ as @CyclotomicField commented, we could derive:$$ J\mathrm{d}\mathbf{S}=\mathbf{F}^\mathsf{T}\mathrm{d}\mathbf{s}, $$ and therefore $$\mathrm{d}\mathbf{s}=\mathbf{F}^{-\mathsf{T}}J\mathrm{d}\mathbf{S}.\tag{33}$$ Note that one can always change the order of inverse and transpose for an invertible matrix.