Transformations

A description of vector transformations and how they're preformed

We can transform a vector to change it into another vector. In general, this vector can be in the same or in a different vector space - for example, a vector along one plane could transform into a vector along another plane (Das, R ).

Fun transformation visualization tool

More information about transformations

Notation

Transformations are labeled with a letter wearing a little hat - like this: T^\hat{T}. To apply the transformation, we multiply the vector by the transformation.

So, the expression...

T^∣x⟩=∣y⟩\hat{T}|x\rangle = |y\rangle

...describes ∣y⟩|y\rangle, which is a result of ∣x⟩|x\rangletransformed by (multiplied by the matrix represented by) T^\hat{T}.

Transformation Matrices

In general, we can use a matrix to define the actions preformed which result in a given transformation. We multiply a vector by the transformation to apply these operations in the proper order.

Examples

The Identity Matrix

LetT^=[100010001],∣u⟩=[357]Let \hspace{8pt} \hat{T}=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}, |u\rangle = \begin{bmatrix}3\\5\\7\end{bmatrix}
T^∣u⟩=[100010001]∗[357]=[357]\hat{T}|u\rangle=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}*\begin{bmatrix}3\\5\\7\end{bmatrix}=\begin{bmatrix}3\\5\\7\end{bmatrix}

If you preform the multiplication here you'll see that this transformation didn't do anything - we call this the identity transformation because it gives us the identity of the matrix we transform it by. This special matrix will come in handy later.

Rotational Matrix

LetR^=[0−110],∣v⟩=[35]Let \hspace{8pt} \hat{R}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}, |v\rangle = \begin{bmatrix}3\\5\end{bmatrix}
R^∣v⟩=[0−110]∗[35]=[−53]\hat{R}|v\rangle=\begin{bmatrix}0&-1\\1&0\end{bmatrix} * \begin{bmatrix}3\\5\end{bmatrix} =\begin{bmatrix}-5\\3\end{bmatrix}

This transformation rotated ∣v⟩|v\rangle by 90∘90^\circ.

Transformation matrix problems

Types of Transformations

Transformations - Generally

In general, any matrix which can be "applied" or multiplied to another matrix can be a transformation.

In the case of a column vector, we're affecting some kind of change on it, taking one thing and "making" it into another. The old vector and the new vector share a particular relationship, which is described by the transformation matrix.

Linear Transformations

A linear transformation is a transformation that affects a vector within the context of vector spaces.

For example, we take a vector along one plane and transform it to a vector along another plane. Or perhaps we take a vector and rotate it within the same plane. We're talking about transformations within the context of linear relationships (Nykam, D ).

More specific details about linear transformations and how they work

Operators

Operators are special linear transformations that transform a vector within it's vector space.

In this case we can't change a vector in one plane into a vector in another plane, but we can rotate a vector along the same plane. This means all operators are linear transformations, but not vice versa.

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