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The linear function
Matrices and linear transformations
A description of how every matrix can be associated with a linear transformation.
A description of how we denote functions.
Definition of a linear transformation of Euclidean spaces.
The definition of differentiability in higher dimensions
The definition of differentiability for multivariable functions. Informal derivation designed to give intuition behind the condition for a function to be differentiable.
Subtleties of differentiability in higher dimensions
A description of some of the tricky ways where a function of multiple variables can fail to be differentiable. Example two variable functions are illustrated with interactive graphics.
The derivative matrix
The derivative matrix is presented as a natural generalization of the single variable derivative to multivariable functions.
Surfaces as graphs of functions
Illustration of how the graph of a scalar-valued function of two variables is a surface.
Determinants and linear transformations
A description of how a determinant describes the geometric properties of a linear transformation.
Geometric properties of the determinant
Explanation of some of the basic geometric properties of the determinant.
Domain of functions
How to determine the domain of a function.
How linear transformations map parallelograms and parallelepipeds
Why linear transformations map parallelograms onto parallelograms and parallelepipeds onto parallelepipeds.
The function machine
The function machine metaphor helps explain the definition and properties of a function.
Basic examples of functions illustrating the definition of a function.
Function machine parameters
Illustrating function parameters as dials that influence the behavior of a function machine.
Function machine composition
Composing two functions simply means connecting together their respective function machines.
The function machine inverse
The inverse of a function is represented by running its function machine backwards.
Function composition examples
Examples of composing functions to form a new combined function.
Inverse function examples
Examples of forming an inverse function that does undoes the action of another function.
The exponential function
Overview of the exponential function and a few of its properties.
Calculating the derivative of a linear function using the derivative formula
Exploring how the limit definition of the derivative gives the slope of a linear function.
The linear function