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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.

The Jacobian matrix is a 2x2 array of first-order partial derivatives. Its rows correspond to the output components $f_1$ and $f_2$, and its columns correspond to the input variables x and y.

Conditions: The function is a vector-valued map from $R^2$ to $R^2$.; The output components are $f_1(x,y) = x + \sin (y)$ and $f_2(x,y) = y + \sin (x)$.; The matrix is evaluated at a specific point, such as (-2, 1), to describe local linear behavior.

The Jacobian matrix is a 2x2 array of first-order partial derivatives. Its rows correspond to the output components $f_1$ and $f_2$, and its columns correspond to the input variables x and y.

Conditions: The function is a vector-valued map from $R^2$ to $R^2$.; The output components are $f_1(x,y) = x + \sin (y)$ and $f_2(x,y) = y + \sin (x)$.; The matrix is evaluated at a specific point, such as (-2, 1), to describe local linear behavior.