Why are the columns of the Jacobian matrix constructed from basis vectors?

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.

Reasoning, step by step

1. Recall the linear algebra rule that matrix columns are images of basis vectors.
2. Apply this rule to the local linear approximation represented by the Jacobian.
3. Identify the first column as the transformed x-basis vector (partial derivatives with respect to x).
4. Identify the second column as the transformed y-basis vector (partial derivatives with respect to y).
5. Assemble the Jacobian matrix column by column based on these images.

Example

The speaker explains that when representing a linear transformation with a matrix, the first column tells where the first basis vector goes and the second column shows where the second basis vector goes. This justifies filling the Jacobian columns with partial derivatives corresponding to x and y steps.

Common misconceptions

• Thinking the rows represent basis vectors instead of columns.
• Assuming the Jacobian columns represent input vectors rather than their transformed outputs.
• Confusing the order of basis vectors; typically column 1 is x and column 2 is y.

Watch the explanation

Explore next

• What does the first column of the Jacobian matrix represent?
• How do you determine the second column of the Jacobian matrix?
• Why is the Jacobian matrix a 2x2 matrix for this function?