Editing Linear map (section)

==Matrices==
{{Main|Transformation matrix}}

If <math>V</math> and <math>W</math> are [[finite-dimensional]] vector spaces and a [[basis of a vector space|basis]] is defined for each vector space, then every linear map from <math>V</math> to <math>W</math> can be represented by a [[matrix (mathematics)|matrix]].<ref>{{harvnb|Rudin|1976|page=210}}

Suppose <math display="inline">\left\{\mathbf{x}_1, \ldots, \mathbf{x}_n\right\}</math> and <math display="inline">\left\{\mathbf{y}_1, \ldots, \mathbf{y}_m\right\}</math> are bases of vector spaces {{mvar|X}} and {{mvar|Y}}, respectively. Then every <math display="inline">A \in L(X, Y)</math> determines a set of numbers <math display="inline">a_{i,j}</math> such that
<math display="block">A\mathbf{x}_j = \sum_{i=1}^m a_{i,j}\mathbf{y}_i\quad (1 \leq j \leq n).</math>

It is convenient to represent these numbers in a rectangular array of {{mvar|m}} rows and {{mvar|n}} columns, called an {{mvar|m}} ''by'' {{mvar|n}} ''matrix'':
<math display="block">[A] = \begin{bmatrix}
  a_{1,1} & a_{1,2} & \ldots & a_{1,n} \\
  a_{2,1} & a_{2,2} & \ldots & a_{2,n} \\ 
   \vdots &  \vdots & \ddots &  \vdots \\
  a_{m,1} & a_{m,2} & \ldots & a_{m,n}
\end{bmatrix}</math>

Observe that the coordinates <math display="inline">a_{i,j}</math> of the vector <math display="inline">A\mathbf{x}_j</math> (with respect to the basis <math display="inline">\{\mathbf{y}_1, \ldots, \mathbf{y}_m\}</math>) appear in the ''j''<sup>th</sup> column of <math display="inline">[A]</math>. The vectors <math display="inline">A\mathbf{x}_j</math> are therefore sometimes called the ''column vectors'' of <math display="inline">[A]</math>. With this terminology, the ''range'' of {{mvar|A}} ''is spanned by the column vectors of <math display="inline">[A]</math>''.
</ref> This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if <math>A</math> is a real <math>m \times n</math> matrix, then <math>f(\mathbf x) = A \mathbf x</math> describes a linear map <math>\R^n \to \R^m</math> (see [[Euclidean space]]).

Let <math>\{ \mathbf {v}_1, \ldots , \mathbf {v}_n \}</math> be a basis for <math>V</math>. Then every vector <math>\mathbf {v} \in V</math> is uniquely determined by the coefficients <math>c_1, \ldots , c_n</math> in the field <math>\R</math>:
<math display="block">\mathbf{v} = c_1 \mathbf{v}_1 + \cdots + c_n \mathbf {v}_n.</math>

If <math display="inline">f: V \to W</math> is a linear map,
<math display="block">f(\mathbf{v}) = f(c_1 \mathbf{v}_1 + \cdots + c_n \mathbf{v}_n) = c_1 f(\mathbf{v}_1) + \cdots + c_n f\left(\mathbf{v}_n\right),</math>

which implies that the function ''f'' is entirely determined by the vectors <math>f(\mathbf {v}_1), \ldots , f(\mathbf {v}_n)</math>. Now let <math>\{ \mathbf {w}_1, \ldots , \mathbf {w}_m \}</math> be a basis for <math>W</math>.  Then we can represent each vector <math>f(\mathbf {v}_j)</math> as
<math display="block">f\left(\mathbf{v}_j\right) = a_{1j} \mathbf{w}_1 + \cdots + a_{mj} \mathbf{w}_m.</math>

Thus, the function <math>f</math> is entirely determined by the values of <math>a_{ij}</math>. If we put these values into an <math>m \times n</math> matrix <math>M</math>, then we can conveniently use it to compute the vector output of <math>f</math> for any vector in <math>V</math>.  To get <math>M</math>, every column <math>j</math> of <math>M</math> is a vector
<math display="block">\begin{pmatrix} a_{1j} \\ \vdots \\ a_{mj} \end{pmatrix}</math>
corresponding to <math>f(\mathbf {v}_j)</math> as defined above. To define it more clearly, for some column <math>j</math> that corresponds to the mapping <math>f(\mathbf {v}_j)</math>,
<math display="block">\mathbf{M} = \begin{pmatrix}
  \ \cdots & a_{1j} & \cdots\  \\
           & \vdots &          \\
           & a_{mj} &
\end{pmatrix}</math>
where <math>M</math> is the matrix of <math>f</math>. In other words, every column <math>j = 1, \ldots, n</math> has a corresponding vector <math>f(\mathbf {v}_j)</math> whose coordinates <math>a_{1j}, \cdots, a_{mj}</math> are the elements of column <math>j</math>. A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen.

The matrices of a linear transformation can be represented visually:

# Matrix for <math display="inline">T</math> relative to <math display="inline">B</math>: <math display="inline">A</math>
# Matrix for <math display="inline">T</math> relative to <math display="inline">B'</math>: <math display="inline">A'</math>
# Transition matrix from <math display="inline">B'</math> to <math display="inline">B</math>: <math display="inline">P</math>
# Transition matrix from <math display="inline">B</math> to <math display="inline">B'</math>: <math display="inline">P^{-1}</math>

[[File:Linear_transformation_visualization.svg|frame|The relationship between matrices in a linear transformation|none]]

Such that starting in the bottom left corner <math display="inline">\left[\mathbf{v}\right]_{B'}</math> and looking for the bottom right corner <math display="inline">\left[T\left(\mathbf{v}\right)\right]_{B'}</math>, one would left-multiply—that is, <math display="inline">A'\left[\mathbf{v}\right]_{B'} = \left[T\left(\mathbf{v}\right)\right]_{B'}</math>. The equivalent method would be the "longer" method going clockwise from the same point such that <math display="inline">\left[\mathbf{v}\right]_{B'}</math> is left-multiplied with <math display="inline">P^{-1}AP</math>, or <math display="inline">P^{-1}AP\left[\mathbf{v}\right]_{B'} = \left[T\left(\mathbf{v}\right)\right]_{B'}</math>.

===Examples in two dimensions===
In two-[[dimension]]al space '''R'''<sup>2</sup> linear maps are described by 2 × 2 [[matrix (mathematics)|matrices]]. These are some examples:

* [[Rotation (mathematics)|rotation]]
** by 90 degrees counterclockwise: <math display="block">\mathbf{A} = \begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}</math>
** by an angle ''θ'' counterclockwise: <math display="block">\mathbf{A} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}</math>
* [[Reflection (mathematics)|reflection]]
** through the ''x'' axis: <math display="block">\mathbf{A} = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}</math>
** through the ''y'' axis: <math display="block">\mathbf{A} = \begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}</math>
** through a line making an angle ''θ'' with the origin: <math display="block">\mathbf{A} = \begin{pmatrix}\cos2\theta & \sin2\theta \\ \sin2\theta & -\cos2\theta \end{pmatrix}</math> 
* [[Scaling (geometry)|scaling]] by 2 in all directions: <math display="block">\mathbf{A} = \begin{pmatrix} 2 & 0\\ 0 & 2\end{pmatrix} = 2\mathbf{I}</math>
* [[shear mapping|horizontal shear mapping]]: <math display="block">\mathbf{A} = \begin{pmatrix} 1 & m\\ 0 & 1\end{pmatrix}</math>
* skew of the ''y'' axis by an angle ''θ'': <math display="block">\mathbf{A} = \begin{pmatrix} 1 & -\sin\theta\\ 0 & \cos\theta\end{pmatrix}</math>
* [[squeeze mapping]]: <math display="block">\mathbf{A} = \begin{pmatrix} k & 0\\ 0 & \frac{1}{k}\end{pmatrix}</math>
* [[Projection (linear algebra)|projection]] onto the ''y'' axis: <math display="block">\mathbf{A} = \begin{pmatrix} 0 & 0\\ 0 & 1\end{pmatrix}.</math>

If a linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a [[conformal linear transformation]].