How does the first column of the Jacobian matrix relate to a tiny step in the x direction?

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.

Reasoning, step by step

1. Consider a tiny step dx in the input space along the x-axis.
2. Apply the nonlinear transformation to this tiny step.
3. Observe the resulting displacement vector in the output space.
4. Identify the horizontal component of this output displacement as ∂$f_1$/∂x.
5. Identify the vertical component of this output displacement as ∂$f_2$/∂x.
6. Place these two components into the first column of the Jacobian matrix.

Example

The video shows a green arrow representing the transformed ∂x step pointing diagonally down and to the right. The speaker explains that its x-component is ∂$f_1$/∂x and its y-component is ∂$f_2$/∂x, which are written into the first column.

Common misconceptions

• Assuming an x-direction input step only affects the first output component.
• Thinking the first column represents the input basis vector itself rather than its transformed image.
• Confusing the raw infinitesimal change with the derivative rate without dividing by the step size.

Watch the explanation

Explore next

• What does the second column of the Jacobian matrix represent?
• Why are matrix columns considered images of basis vectors?
• How does the Jacobian matrix capture local linearity?