[?] Use transformation matrices that act on vector?

Questions
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1

Hi ,
I’m new with Mathematica and would like to know how to write transformation matrices (operators) that act on vectors (for tensor analysis). Or a Jacobinan-type matrix For example, a differential operator that acts on a column vector of functions:
row 1: d[ ]/dx1 d[ ]/dx2
row 2: d[ ]/dx3 d[ ]/dx4

How can I do that in Mathematica? The regular linear algebra functions don’t work.
AUSP6

2

For the Jacobian matrix you can use the derivative operator D with the following syntax:

vector = {x, x*z, x^2 - y};
variables = {x, y, z};
jacobianMatrix = D[vector, {variables}];
MatrixForm[jacobianMatrix]
3

Have you tried the Wolfram Language documentation?

Linear algebra