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Application: An important part in many applications such as optimization, mechanics, and partial differential equations (PDE) is taking derivatives with respect to vectors. By definition, if y = [y1, y2, ..., yn] and x = [x1, x2, ..., xn] are two vectors and y = f(x), which means y is dependent on x, then dyi/dxj = Dyi/Dxj.
1. Using the definition, determine if y = Ax when A = [b2, A].
Make a guess: What if A is a general n x n matrix?
2. The definition also holds even when y is a scalar or a vector of length 1. What is dy/dx?
Hint: You can try with x having length 2, 3, and generalize.