Determining size of Jacobian Matrix of Function Composition

Question

Say I have two functions:

$ f: R^{2} \rightarrow R^{8} $

$ g: R^{8} \rightarrow R^{4} $

And I want to compute the Jacobian matrix of the composition $ f \circ g $:

$ J_{f \circ g} $

What will the size of this Jacobian matrix be? Will it be 2x4?

Answer 1

Note that the composition $f \circ g$ is meaningless.

We can consider $g \circ f: R^{2} \rightarrow R^{4}$ and in that case we have

• $J_f$ is $8\times 2$

• $J_g$ is $4\times 8$

then

• $J_{g \circ f}=J_g\cdot J_f$ which is $4\times 2$

Answer 2

The composition $f \circ g $ is not defined, since $g (x) \notin R^2$.