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1. Dual space

A linear function on an euclidean space

f(x1, x2, ..., xn) = a1x1 + a2x2 + ... + anxn

is simply the left side of the usual linear equation. Note that adding and scalar multiplying such linear functions still produce linear functions. Therefore such linear functions form a vector space. More generally, we have

A linear function on a vector space V is f: VR satisfying

f(u + v) = f(u) + f(v), f(cu) = c f(u).

The dual space V* is the collection of all linear functions.

Using the concept of Hom space, we have

V* = Hom(V, R).

In particular, V* is a vector space with the following addition and scalar multiplication

(f + g)(v) = f(v) + g(v), (cf)(v) = c( f(v)).

The definition above must be justified by showing that a linear function f: RnR is given by the familier formula. Write a vector of Rn as

(x1, x2, ..., xn) = x1e1 + x2e2 + ... + xnen,

where {e1, e2, ..., en} is the standard basis of Rn. Then the linearity of f implies

f(x1, x2, ..., xn)
= f(x1e1 + x2e2 + ... + xnen)
= x1f(e1) + x2f(e2) + ... + xnf(en)
= a1x1 + a2x2 + ... + anxn,

where ai = f(ei). Thus we conclude that

(Rn)* = {f(x1, x2, ..., xn) = a1x1 + a2x2 + ... + anxn: a1, a2, ..., an are real numbers}.

Note that the coefficients of the linear function gives an identification

ιn: (Rn)*Rn, a1x1 + a2x2 + ... + anxn ↔ (a1, a2, ..., an).

The argument above is a special case of the proof that linear transformations between euclidean spaces are given by matrices. In fact, as a linear transformation from Rn to R1, the function f is given by the 1 by n matrix [f(e1) f(e2) ... f(en)].

Example For a linear function f: P3R, we have

f(c0 + c1t + c2t2 + c3t3) = c0f(1) + c1f(t) + c2f(t2) + c3f(t)3 = c0a0 + c1a1 + c2a2 + c3a3.


(P3)* = {f(c0 + c1t + c2t2 + c3t3) = c0a0 + c1a1 + c2a2 + c3a3: a0, a1, a2, a3 are real numbers}.

We actually see a pattern here. Basically, linear functions on (finite dimensional) vector spaces are determined by the values on basis vectors. For example, we also have

(M(2, 2))* = {f [ x y ] = ax + by + cz + dw: a, b, c, d are real numbers}.
z w

Example In this example, we saw the evaluation

ff(2): F(R) → R

at t = 2 is a linear function. Clearly the evaluations at the other places and the linear combinations of evaluations such as

f → 2f(2) - 3f(1): F(R) → R

is also linear.

Another linear function on function spaces is given by the integration (see this exercise)

f → ∫01f(t) dt: C(R) → R.

Integration after multiplying a function such as

f → ∫01f(t)cosπt dt: C(R) → R

is also a linear function.

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