Composition of Linear Mappings
Any real m × n matrix A gives rise to a linear transformation ℝn → ℝm which maps each vector x→ in ℝn to the matrix-vector product A⋅x→, which is a vector in ℝm.
Conversely, each linear mapping L:ℝn → ℝm can be represented by a unique m × n transformation matrix A, where the i, j-entry of A is the ith coordinate of Lej and ej is the standard basis vector with 1 in the jth 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 × m matrix B represents a linear mapping g: ℝm → ℝk, and a real m × n matrix A represents a linear mapping f:ℝn → ℝm, then the composition g∘f: ℝn → ℝk is represented by the matrix BA.
Proof: g∘fx→ = gfx→ = gA⋅x→ = B⋅A⋅x→ = BA⋅x→ due to the associativity of matrix multiplication.
Illustrating Common Linear Transformations
To graphically illustrate linear mappings in two-dimensional space (ℝ2), we can use 2 × 2 real matrices to describe transformations of the unit square, which is defined by the columns of the identity matrix:1001.
The matrix A = acbd can be viewed as the transformation of the unit square into a parallelogram with vertices at 0,0, a,b, a+c,b+d , and c,d.
This allows certain types of matrices to represent common transformations of planar figures, including:
Reflection through the y-axis, A = −1001
Reflection through the x-axis, A = 100−1
Rotation by θ degrees counter-clockwise, A = cos(θ)−sin(θ)sin(θ)cos(θ)
Scaling by a factor of k in all directions, A = k00k
Horizontal shear, A = 1m01
Squeeze mapping, A = r001r
Projection onto the y-axis, A = 0001
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.
Change the values in matrices A and B below to observe how the composition of the corresponding linear mappings transforms the unit square. Click the checkboxes to highlight the different linear transformations.
The plot on the left illustrates the parallelograms which result from the transformation by matrix A, the transformation by matrix B and the successive transformations with A followed by B.
The plot on the right illustrates the parallelogram which results from the transformation by matrix BA. This figure will always match the parallelogram in the other plot which was created by the composition of linear transformations A then B.
If either matrix A or B is left as the identity matrix, you can observe the transformation of the unit square caused by a single linear mapping.
In both of the plots below, the gray figure (show
) is the unit square defined by the standard basis vectors: 10 and 01.
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.
The purple parallelogram represents the transformation of the unit square by matrix BA.
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