3 Linear function - Auburn University
so property (b) holds. Therefore, L is linear. If a is any number, then the function f : R→ Rgiven by f(x) = ax has as its graph a straight line (through the origin with slope a). In fact, this function is linear in the sense of the above definition (regarding R as the same thing as R1). The next theorem generalizes this statement with the ...
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22 Derivative of inverse function - Auburn University
web.auburn.edu22.2 Derivative of logarithm function The logarithm function log a xis the inverse of the exponential function ax. Therefore, we can use the formula from the previous section to obtain its deriva-tive. Derivative of logarithm function. For any positive real number a, d dx [log a x] = 1 xlna: In particular, d dx [lnx] = 1 x:
9 Inner product - Auburn University
web.auburn.eduIt is certain properties of the dot product that make this work. 9 INNERPRODUCT 3 The generalization of the dot product to an arbitrary vector space is called an “inner product.” Just like the dot product, this is a certain way of putting two vectors together to get a number. The properties it satisfies are enough to
4 Span and subspace - Auburn University
web.auburn.eduSubspace. A subset S of Rn is called a subspaceif the following hold: (a) 0∈ S, (b) x,y∈ S implies x+y∈ S, (c) x∈ S,α ∈ Rimplies αx∈ S. In other words, a subset S of Rn is a subspace if it satisfies the following: (a) S contains the origin 0, (b) S is closed under addition (meaning, if xand yare two vectors in S, then
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