### Dual

##### 1. Dual space

A linear function on an euclidean space

`f`(`x`_{1}, `x`_{2}, ..., `x`_{n}) = `a`_{1}`x`_{1} + `a`_{2}`x`_{2} + ... + `a`_{n}x_{n}

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`: `V` → **R** satisfying

`f`(`u` + `v`) = `f`(`u`) + `f`(`v`), `f`(`c`**u**) = `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`: **R**^{n} → **R** is given by the familier formula. Write a vector of **R**^{n} as

(`x`_{1}, `x`_{2}, ..., `x`_{n}) = `x`_{1}**e**_{1} + `x`_{2}**e**_{2} + ... + `x`_{n}**e**_{n},

where {**e**_{1}, **e**_{2}, ..., **e**_{n}} is the standard basis of **R**^{n}. Then the linearity of `f` implies

`f`(`x`_{1}, `x`_{2}, ..., `x`_{n})

=
`f`(`x`_{1}**e**_{1} + `x`_{2}**e**_{2} + ... + `x`_{n}**e**_{n})

=
`x`_{1}`f`(**e**_{1}) + `x`_{2}`f`(**e**_{2}) + ... + `x`_{n}f(**e**_{n})

=
`a`_{1}`x`_{1} + `a`_{2}`x`_{2} + ... + `a`_{n}x_{n},

where `a`_{i} = `f`(**e**_{i}). Thus we conclude that

(**R**^{n})^{*} = {`f`(`x`_{1}, `x`_{2}, ..., `x`_{n}) = `a`_{1}`x`_{1} + `a`_{2}`x`_{2} + ... + `a`_{n}x_{n}: `a`_{1}, `a`_{2}, ..., `a`_{n} are real numbers}.

Note that the coefficients of the linear function gives an identification

`ι`_{n}: (**R**^{n})^{*} ↔ **R**^{n},
`a`_{1}`x`_{1} + `a`_{2}`x`_{2} + ... + `a`_{n}x_{n}
↔ (`a`_{1}, `a`_{2}, ..., `a`_{n}).

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 **R**^{n} to **R**^{1}, the function *f* is given by the 1 by `n` matrix [`f`(**e**_{1}) `f`(**e**_{2}) ... `f`(**e**_{n})].

Example For a linear function `f`: `P`_{3} → **R**, we have

`f`(`c`_{0} + `c`_{1}`t` + `c`_{2}`t`^{2} + `c`_{3}`t`^{3}) = `c`_{0}`f`(1) + `c`_{1}`f`(`t`) + `c`_{2}`f`(`t`^{2}) + `c`_{3}`f`(`t`)^{3} = `c`_{0}`a`_{0} + `c`_{1}`a`_{1} + `c`_{2}`a`_{2} + `c`_{3}`a`_{3}.

Therefore

(`P`_{3})^{*} = {`f`(`c`_{0} + `c`_{1}`t` + `c`_{2}`t`^{2} + `c`_{3}`t`^{3}) = `c`_{0}`a`_{0} + `c`_{1}`a`_{1} + `c`_{2}`a`_{2} + `c`_{3}`a`_{3}: `a`_{0}, `a`_{1}, `a`_{2}, `a`_{3} 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

`f` → `f`(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` → 2`f`(2) - 3`f`(1): `F`(**R**) → **R**

is also linear.

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

`f` → ∫_{0}^{1}`f`(`t`) `dt`: `C`(**R**) → **R**.

Integration after multiplying a function such as

`f` → ∫_{0}^{1}`f`(`t`)cos`πt` `dt`: `C`(**R**) → **R**

is also a linear function.