Finding a Composite Transformation from Geometric Actions

From a Sequence of Actions to One Matrix

A geometric description often specifies a transformation as a sequence: rotate a vector, reflect the result, then apply a shear. To calculate the final vector efficiently, translate each action into its standard matrix and combine those matrices. The essential bookkeeping is the order: with column vectors, the matrix for the first action appears on the right of the product, because it acts first.

Here we work in \(\mathbb{R}^2\). All vectors are columns, and the rotations, reflections, projections, and shears considered are linear transformations that fix the origin. A translation, for example, is not one of these linear transformations. The standard matrix of a linear transformation \(T:\mathbb{R}^2\to\mathbb{R}^2\) is the matrix \(\mathbf{A}\) satisfying \(T(\mathbf{x})=\mathbf{A}\mathbf{x}\) for every \(\mathbf{x}\in\mathbb{R}^2\).

Some common matrices in \(\mathbb{R}^2\) are listed below. In the rotation matrix, \(\theta\) is measured counterclockwise from the positive \(x\)-axis. The projection and reflection shown are onto and across the indicated lines through the origin. A horizontal shear adds \(k\) times the \(y\)-coordinate to the \(x\)-coordinate; a vertical shear adds \(k\) times the \(x\)-coordinate to the \(y\)-coordinate.

These matrices follow from the images of the standard basis vectors \(\mathbf{e}_1=(1,0)\) and \(\mathbf{e}_2=(0,1)\): the first column is the image of \(\mathbf{e}_1\), and the second column is the image of \(\mathbf{e}_2\). For instance, a horizontal shear sends \((x,y)\) to \((x+ky,y)\), so it sends \(\mathbf{e}_1\) to \((1,0)\) and \(\mathbf{e}_2\) to \((k,1)\). The columns give the displayed matrix.

Why the Product Has This Order

Proof. By the definition of standard matrix, \(T_1(\mathbf{x})=\mathbf{A}_1\mathbf{x}\). Applying \(T_2\) to that output gives \(T_2(T_1(\mathbf{x}))=\mathbf{A}_2(\mathbf{A}_1\mathbf{x})=(\mathbf{A}_2\mathbf{A}_1)\mathbf{x}\). Continuing one action at a time, after \(j\) actions the output is \((\mathbf{A}_j\cdots\mathbf{A}_1)\mathbf{x}\): this holds for \(j=1\), and if it holds for \(j\), applying \(T_{j+1}\) multiplies the output on the left by \(\mathbf{A}_{j+1}\), giving \((\mathbf{A}_{j+1}\mathbf{A}_j\cdots\mathbf{A}_1)\mathbf{x}\). Taking \(j=k\) proves the formula. \(\square\)

The theorem also gives a practical procedure: write the actions in the order they happen, put their matrices in reverse reading order in the product, and apply the resulting product to the input column. The matrix product itself can then be checked by applying the individual actions to the same input.

Order Can Change the Result

Matrix multiplication records the order of action; it is not generally valid to rearrange the factors. This is especially visible when one action discards information, as a projection does. In the next example, both sequences use the same two transformations, but they do not have the same effect.

For a fixed input, two different matrices might still happen to give the same output. To say that two composites agree for every input requires their matrices to agree, as the next result shows.

Proof. If \(\mathbf{A}=\mathbf{B}\), then for every \(\mathbf{x}\), \(T(\mathbf{x})=\mathbf{A}\mathbf{x}=\mathbf{B}\mathbf{x}=U(\mathbf{x})\). Conversely, suppose \(T(\mathbf{x})=U(\mathbf{x})\) for every \(\mathbf{x}\). In particular this holds for \(\mathbf{e}_1\) and \(\mathbf{e}_2\). The first columns of \(\mathbf{A}\) and \(\mathbf{B}\) are \(T(\mathbf{e}_1)\) and \(U(\mathbf{e}_1)\), so they are equal. Their second columns are \(T(\mathbf{e}_2)\) and \(U(\mathbf{e}_2)\), so they are equal as well. Both columns agree, which gives \(\mathbf{A}=\mathbf{B}\). \(\square\)

Three Actions, One Composite Matrix

A Reliable Checklist and a Common Mistake

When a problem describes a sequence of geometric actions, use the following steps:

1. Write down the actions in the order they happen.
2. Choose a standard matrix for each action, using the stated angle, axis, or shear factor.
3. Write the matrices in reverse order in the product, with the first action at the far right.
4. Multiply the matrices in manageable pairs, showing each entry of the products.
5. Apply the final matrix to the input column and check the result by doing the actions one at a time.

Common mistake: multiplying in the order the actions are spoken. In the first worked example, writing \(\mathbf{R}\mathbf{F}\mathbf{H}\) instead of \(\mathbf{H}\mathbf{F}\mathbf{R}\) reverses the sequence. For the input \(\begin{bmatrix}1\\2\end{bmatrix}\), this incorrect product acts as follows: the shear gives \(\begin{bmatrix}5\\2\end{bmatrix}\), the reflection gives \(\begin{bmatrix}5\\-2\end{bmatrix}\), and the rotation gives \(\begin{bmatrix}2\\5\end{bmatrix}\). But the specified sequence gives \(\begin{bmatrix}-4\\-1\end{bmatrix}\), as checked above. The correction is to place the first action's matrix at the right, so the vector meets that matrix first.

A second useful check is dimensional: each matrix must be \(2\times2\) here, and the product must act on a \(2\times1\) input column. More importantly, verifying a product on one input confirms that calculation for that input, not necessarily for all inputs. To establish that two composites are the same transformation, compare their matrices, or compare their effects on both standard basis vectors.