Dual vector space

Definition

Given a vector space $(V,F)$ we define the dual or conjugate vector space (which we denote $V^*$) as:

• $$V^*=\text{Hom}(V,F)$$ (recall this the set of all homomorphisms (specifically linear ones) from $V$ to $F$)
• That is $$V^*=\{f:V\rightarrow F|\ f\text{ is linear}\}$$ (which is to say $$f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)\ \forall x,y\in V\ \forall \alpha,\beta\in F$$ )

We usually denote the elements of $V^*$ with $*$s after them, that is something like $$f^*\in V^*$$

We call the elements of $V^*$:

• Covectors
• Dual vectors
• Linear form
• Linear functional (and $V^*$ the set of linear functionals)

Covectors and Dual vectors are interchangeable and I find myself using both without a second thought.

Equality of covectors

We say two covectors, $$f^*,g^*:V\rightarrow F$$ are equal if:

• $$\forall v\in V[f^*(v)=g^*(v)]$$ - that is they agree on their domain (as is usual for function equality)

The zero covector

The zero covector $$:V\rightarrow F$$ with $$v\mapsto 0$$ {{Todo|Confirm it is written $0^*$ before writing that here

Examples

Dual vectors of $\mathbb{R}^2$

Consider (for $v\in\mathbb{R}^2$: $$f^*(v)=2x$$ and $$g^*(v)=y-x$$ - it is easy to see these are linear and thus are covectors!

Dual vectors of $\mathbb{R}[x]_{\le 2}$

(Recall that $$\mathbb{R}[x]_{\le 2}$$ denotes all polynomials up to order 2, that is: $$\alpha+\beta x+\gamma x^2$$ for $$\alpha,\beta,\gamma\in\mathbb{R}$$ in this case)

Consider $p\in\mathbb{R}[x]_{\le 2}$ and $$f*,g^*:\mathbb{R}[x]_{\le 2}\rightarrow\mathbb{R}$$ given by:

• $$f^*(p)=\int^{+\infty}_0e^{-x}p(x)dx$$ - this is a covector

  To see this notice: $$f^*(\alpha p+\beta q)=\int^{+\infty}_0e^{-x}[\alpha p(x)+\beta q(x)]dx$$ $$=\alpha\int^{+\infty}_0e^{-x}p(x)dx+\beta\int^{+\infty}_0e^{-x}q(x)dx$$

• $$g^*(p)=\frac{dp}{dx}\Big|_{x=1}$$ (note that: $$g^*(\alpha+\beta x+\gamma x^2)=\beta+2\gamma$$ - so this isn't just projecting to coefficients) is also a covector

Dual basis

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Theorem: Given a basis $\{e_1,\cdots,e_n\}$ of a vector space $(V,F)$ there is a corresponding basis to $V^*$, $\{e_1^*,\cdots,e_n^*\}$ where each $e^*_i$ is the $i^\text{th}$ coordinate of a vector $v\in V$ (that is the coefficient of $e_i$ when $v$ is expressed as $\sum^n_{j=1}v_je_j$). That is to say that $e_i^*$ projects $v$ onto it's $e_i^\text{th}$ coordinate.

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TODO: Proof

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