Main Concept

Any real $m \times n$ matrix A gives rise to a linear transformation ${\mathrm{\ℝ}}^n \to {\mathrm{\ℝ}}^m$ which maps each vector $\overrightarrow{x}$ in ${\mathrm{\ℝ}}^n$ to the matrix-vector product $A\cdot \overrightarrow{x}$, which is a vector in ${\mathrm{\ℝ}}^m$.

Conversely, each linear mapping $L\:{\mathrm{\ℝ}}^n \to {\mathrm{\ℝ}}^m$ can be represented by a unique $m \times n$ transformation matrix A, where the $\left(i\, j\right)$-entry of A is the $i^{\mathrm{th}}$ coordinate of $L\left(e_j\right)$ and $e_j$ is the standard basis vector with 1 in the $j^{\mathrm{th}}$ position and 0 elsewhere.

Due to this 1-1 correspondence between matrices and linear mappings, matrix multiplication corresponds to the composition of linear mappings.

If a real $k \times m$ matrix B represents a linear mapping $g\: {\mathrm{\ℝ}}^m \to {\mathrm{\ℝ}}^k$, and a real $m \times n$ matrix A represents a linear mapping $f\:{\mathrm{\ℝ}}^n \to {\mathrm{\ℝ}}^m$, then the composition $g\circ f\: {\mathrm{\ℝ}}^n \to {\mathrm{\ℝ}}^k$ is represented by the matrix $\mathrm{BA}$.

Proof: $\left(g\circ f\right)\left(\overrightarrow{x}\right) \= g\left(f\left(\overrightarrow{x}\right)\right) \= g\left(A\cdot \overrightarrow{x}\right) \= B\cdot \left(A\cdot \overrightarrow{x}\right) \= \mathrm{BA}\cdot \overrightarrow{x}$ due to the associativity of matrix multiplication.

Illustrating Common Linear Transformations

To graphically illustrate linear mappings in two-dimensional space (${\mathrm{\ℝ}}^2$), we can use $2 \times 2$ real matrices to describe transformations of the unit square, which is defined by the columns of the identity matrix:$\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]$. The matrix $A \= \left[\begin{array}{rr}a& c\\ b& d\end{array}\right]$ can be viewed as the transformation of the unit square into a parallelogram with vertices at $\left(0\,0\right)$, $\left(a\,b\right)$, $\left(a\+c\,b\+d \right)$, and $\left(c\,d\right)$. This allows certain types of matrices to represent common transformations of planar figures, including: •  Reflection through the $y$-axis, $A \= \left[\begin{array}{rr}-1& 0\\ 0& 1\end{array}\right]$ •  Reflection through the $x$-axis, $A \= \left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]$ •  Rotation by θ degrees counter-clockwise, $A \= \left[\begin{array}{rr}\cos \(\mathrm{\θ}\)& -\sin \(\mathrm{\θ}\)\\ \sin \(\mathrm{\θ}\)& \cos \(\mathrm{\θ}\)\end{array}\right]$ •  Scaling by a factor of k in all directions, $A \= \left[\begin{array}{rr}k& 0\\ 0& k\end{array}\right]$ •  Horizontal shear, $A \= \left[\begin{array}{rr}1& m\\ 0& 1\end{array}\right]$ •  Squeeze mapping, $A \= \left[\begin{array}{rr}r& 0\\ 0& \frac{1}{r}\end{array}\right]$ •  Projection onto the $y$-axis, $A \= \left[\begin{array}{rr}0& 0\\ 0& 1\end{array}\right]$   The parallelogram which results from the transformation of the unit square by a matrix BA will be congruent to the parallelogram formed by successive transformations by matrices A and B. If the combination of transformations resulting from matrices A and B returns the original unit square, these matrices and their corresponding linear mappings must be inverses, since their matrix product BA is equal to the identity matrix.

B

A

$\overrightarrow{\mathit{x}}$

$\left(\mathit{B}\mathbf{\circ }\mathit{A}\right)\mathbf{\cdot }\overrightarrow{\mathit{x}}$

BA

$\overrightarrow{\mathit{x}}$

$\mathit{BA}\mathbf{\cdot }\overrightarrow{\mathit{x}}$

$\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

$\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

In both of the plots below, the gray figure (show ) is the unit square defined by the standard basis vectors: $\left[\begin{array}{r}1\\ 0\end{array}\right]$ and $\left[\begin{array}{r}0\\ 1\end{array}\right]$.

In this plot:

  The red parallelogram represents the first transformation of the unit square by matrix A.

The blue parallelogram represents the second transformation of this figure by matrix B.

The purple parallelogram represents the composition of transformations by matrices A then B.

In this plot:

The purple parallelogram represents the transformation of the unit square by matrix BA.

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