# Linear forms and the dual space, including the Orthogonal group, special orthogonal group, special linear group

## Linear forms

### Linear forms

A linear form is a linear map from a vector space to a scalar from the vector space’s underlying field.

$$\hom (V, F)$$

#### Matrix operators

Linear forms can be represented as matrix operators.

$$v^TM=f$$

Where $$M$$ has only one column.

#### Stuff

$$f(M)=f(v)$$

We introduce $$e_i$$, the element vector. This is $$0$$ for all entries except for $$i$$ where it is $$1$$. Any vector can be shown as a sum of these vectors multiplied by a scalar.

$$f(M)=f(\sum^m_{i=1}a_{i}e_i)$$

$$f(M)=\sum_{i=1}^mf(a_{i}e_i)$$

$$f(M)=\sum_{i=1}^ma_if(e_i)$$

$$f(M)=\sum_{i=1}^ma_if(e_i)$$

#### Orthonormal basis

$$f(M)=\sum_{i=1}^ma_i$$

### Dual space

The dual space $$V^*$$ of vector space $$V$$ is the set of all linear forms, $$\hom(V,F)$$.

#### The dual space is itself a vector space

$$v\in V$$

$$f\in F$$

$$av = f$$

$$bv = g$$

$$(a\oplus b)v=f+g$$

$$(a\oplus b)v=av + bv$$

So there is some operation we can do on two members of dual space

Linear in addition. That is, if we have two dual "things", we can define the addition of functions as the operation which results int he outputs being added.

what about linear in scalar? same approach.

Well we define

$$(c\odot a)=cav$$

### The dual space forms a vector space

The dual space forms a vector space. We can define addition and scalar multiplication on members of the dual space.

The dimension of the dual space is the same as the underlying space.

We have defined the dual space. A vector in dual space will have also have components and a basis.

$$\mathbf w=\sum_i w_i f^j$$

So how we describe the components will depend on the choice of basis.

We choose the dual basis, the basis for $$V^*$$ as:

$$\mathbf e_i \mathbf f^j =\delta_i^j$$

If the basis changes, so does the dual basis.

We write the dual basis as $$e^j$$

## Bilinear forms

### Bilinear forms

A bilinear form takes two vectors and produces a scalar from the underyling field.

This is in contrast to a linear form, which only has one input.

In addition, the function is linear in both arguments.

$$\phi (au+x, bv+y)=\phi (au,bv)+\phi (au,y)+\phi (x,bv)+\phi (x,y)$$

$$\phi (au+x, bv+y)=ab\phi (u,v)+a\phi (u,y)+b\phi (x,v)+\phi (x,y)$$

#### Representing bilinear forms

They can be represented as:

$$\phi (u,v)=v^TMu$$

$$f(M)=f([v_1,v_2])$$

We introduce $$e_i$$, the element vector. This is $$0$$ for all entries except for $$i$$ where it is $$1$$. Any vector can be shown as a sum of these vectors multiplied by a scalar.

$$f(M)=f([\sum^m_{i=1}a_{1i}e_i,\sum^m_{i=1}a_{2i}e_i])$$

$$f(M)=\sum_{k=1}^mf([a_{1k}e_k,\sum^m_{i=1}a_{2i}e_i])$$

$$f(M)=\sum_{k=1}^m\sum^m_{i=1}f([a_{1k}e_k,a_{2i}e_i])$$

Because this in linear in scalars:

$$f(M)=\sum_{k=1}^m\sum^m_{i=1}a_{1k}a_{2i}f([e_k,e_i])$$

$$f(M)=\sum_{k=1}^m\sum^m_{i=1}a_{1k}a_{2i}e_k^TMe_i$$

#### Orthonormal basis and $$M=I$$

$$f(M)=\sum_{k=1}^m\sum^m_{i=1}a_{1k}a_{2i}e_k^Te_i$$

$$f(M)=\sum_{k=1}^m\sum^m_{i=1}a_{1k}a_{2i}\delta_i^k$$

$$f(M)=\sum^m_{i=1}a_{1i}a_{2i}$$

### The dot product

$$v^TMu=f$$

If the operator is $$I$$ then we have the dot product.

$$v^Tu$$

### Orthogonal vectors

Given a metric $$M$$, two vectors $$v$$ and $$u$$ are orthogonal if:

$$v^TMu=0$$

For example if we have the metric $$M=I$$, then two vectors are orthogonal if:

$$v^Tu=0$$

### Metric-preserving transformations and isometry groups

If we have a bilinear form we can write the form as:

$$u^TMv$$

After a transformation $$P$$ to the vectors it is:

$$(Pu)^TM(Pv)$$

$$u^TP^TMPv$$

So the value of the metric will be unaffected if:

$$u^TP^TMPv=u^TMv$$

$$P^TMP=M$$

#### Equivalent metrics

Different metrics can produce the same group. For example multiplying the metric by a constant.

$$P^TMP=M$$

### Orthogonal groups $$O(n, F)$$

#### Recap: Metric-preserving transformations

The bilinear form is:

$$u^TMv$$

The transformations which preserve this are:

$$P^TMP=M$$

#### The orthogonal group

If the metic is $$M=I$$ then the condition is:

$$P^TP=I$$

$$P^T=P^{-1}$$

These form the orthogonal group.

We use $$O$$ instead of $$P$$:

$$O^T=O^{-1}$$

#### Rotations and reflections

The orthogonal group is the rotations and reflections.

#### Parameters of the orthogonal group

The orthogonal group depends on the dimension of the vector space, and the underlying field. So we can have:

• $$O(n, R)$$; and

• $$O(n, C)$$.

#### We generally refer only to the reals

$$O(n)$$ means $$O(n,R)$$.

The generally refer to the reals only.

## Special groups

### Special orthogonal groups $$SO(n, F)$$

The special orthogonal group, $$SO(n,F)$$, is the subgroup of the orthogonal group where $$|M|=1$$.

As a result it includes only the rotation operators, not the flip operators.

$$SO(3)$$ is rotations in 3d space.

$$SO(2)$$ is rotations in 2d space.

#### Determinant of the orthogonal group

The orthogonal group has determinants of $$-1$$ or $$1$$.

$$O^T=O^{-1}$$

$$\det (O^T)=\det (O^{-1})$$

$$\det O=\dfrac{1}{\det O}$$

$$\det O=\pm 1$$

### Special lnear groups $$SL(n, F)$$

The special linear group, $$SL(n,F)$$, is the subgroup of $$GL(n,F)$$ where the determinants are $$1$$.

That is, $$|M|=1$$

These are endomorphisms, not forms.