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The first column of the Jacobian matrix records the output components resulting from an infinitesimal step in the x direction. Specifically, the top entry is the horizontal change in the first output component, and the bottom entry is the vertical change in the second output component, both caused by the x-step.

Conditions: The input perturbation is taken strictly in the positive x direction.; The output is a 2D vector with components $f_1$ and $f_2$.; The step size is infinitesimal, allowing linear approximation.

The first column of the Jacobian matrix records the output components resulting from an infinitesimal step in the x direction. Specifically, the top entry is the horizontal change in the first output component, and the bottom entry is the vertical change in the second output component, both caused by the x-step.

Conditions: The input perturbation is taken strictly in the positive x direction.; The output is a 2D vector with components $f_1$ and $f_2$.; The step size is infinitesimal, allowing linear approximation.

The columns of the Jacobian matrix are constructed from basis vectors because, for any linear transformation, the matrix columns represent the images of the domain's basis vectors. The Jacobian is the matrix of the local linear approximation, so its first column is the image of the x-basis direction and its second column is the image of the y-basis direction.

Conditions: The transformation is being approximated locally by a linear map.; The standard basis vectors (x-direction and y-direction) are used.; The matrix representation follows standard linear algebra rules.

The columns of the Jacobian matrix are constructed from basis vectors because, for any linear transformation, the matrix columns represent the images of the domain's basis vectors. The Jacobian is the matrix of the local linear approximation, so its first column is the image of the x-basis direction and its second column is the image of the y-basis direction.

Conditions: The transformation is being approximated locally by a linear map.; The standard basis vectors (x-direction and y-direction) are used.; The matrix representation follows standard linear algebra rules.