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1.3 What are Linear Functions? 15
1.3 What are Linear Functions?
In calculus classes, the main subject of investigation was the rates of change
of functions. In linear algebra, functions will again be the focus of your
attention, but functions of a very special type. In precalculus you were
perhaps encouraged to think of a function as a machine f into which one
may feed a real number. For each input x this machine outputs a single real
number f(x).
In linear algebra, the functions we study will have vectors (of some type)
as both inputs and outputs. We just saw that vectors are objects that can be
added or scalar multiplied—a very general notion—so the functions we are
going to study will look novel at first. So things don’t get too abstract, here
are five questions that can be rephrased in terms of functions of vectors.
Example 3 (Questions involving Functions of Vectors in Disguise)
(A) What number x satisfies 10x = 3?
⎛ ⎞ ⎛ ⎞
1 0
(B) What 3-vector u satisfies 4 ⎝1⎠ × u = ⎝1⎠?
0 1
(C) What polynomial p satisfies ∫ 1
−1 p(y)dy = 0 and ∫ 1
−1
yp(y)dy = 1?
(D) What power series f(x) satisfies x d
dxf(x) − 2f(x) = 0?
4 The cross product appears in this equation.
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