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This derivation shows: A linear operator L on Rn is completely determined by its action on the standard basis vectors ek . That is, for any vector a ∈ Rn , we can use the sum above to determine L[a] by linearly combining L[e1 ], . . , L[en ]. 10 (Expanding a linear map). 8 given by f (x, y) = (3x, 2x + y, −y). We have f (e1 ) = f (1, 0) = (3, 2, 0) and f (e2 ) = f (0, 1) = (0, 1, −1). Thus, the formula above shows:     3 0 f (x, y) = xf (e1 ) + yf (e2 ) = x  2  + y  1  . 1 Matrices The expansion of linear maps above suggests a context in which it is useful to store multiple vectors in the same structure.

We can differentiate f in any direction v via the directional derivative Dv f : Dv f (x) ≡ d f (x + tv)|t=0 = ∇f (x) · v. dt We allow v to have any length, with the property Dcv f (x) = cDv f (x). 18 (R2 ). Take f (x, y) = x2 y 3 . Then, ∂f = 3x2 y 2 . ∂y ∂f = 2xy 3 ∂x Equivalently, ∇f (x, y) = (2xy 3 , 3x2 y 2 ). So, the derivative of f at (x, y) = (1, 2) in the direction (−1, 4) is given by (−1, 4) · ∇f (1, 2) = (−1, 4) · (16, 12) = 32. There are a few derivatives that we will use many times.

1 Computing Vector Norms . . . . . . . . . . . . . . . . . . . . 2 Larger-Scale Example: Summation . . . . . . . . . . . . . . . 27 28 29 31 32 33 35 36 37 38 umerical analysis introduces a shift from working with ints and longs to floats and doubles. This seemingly innocent transition shatters intuition from integer arithmetic, requiring adjustment of how we must think about basic algorithmic design and implementation. Unlike discrete algorithms, numerical algorithms cannot always yield exact solutions even to well-studied and well-posed problems.

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