Question
A. Explain the general relationship between linear algebra and calculus.

B. Discuss how the definitions of directional derivative (from calculus) and dot product (also called the inner product) can be used to justify the following statement: The gradient points in the direction of maximum increase of a differentiable function from R2 to R.
Solution
A. Explain the general relationship between linear algebra and calculus.

There are many such relationships, that are apparent when we want to do calculus.
The first one is that the definition of derivative of a function of a single variable is defined by seeking for a linear approximation to the function, as defined by its tangent line to the graph.

When we do calculus of more variables the need for linear algebra becomes more evident.
Again when we think of approximating a function of more variables by a best linear function, we are naturally looking for an answer in terms of linear algebra language.

If we want to determine if a function has local maxima or local minima, we need to compute the eigenvalues of the Hessian matrix (a symmetric matrix computed from the second derivatives of a function), or at least say something about these eigenvalues (all negative or all positive), certainly a linear algebra...

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