Shears and Elementary Linear Maps

What a Shear Changes—and What It Leaves Alone

A shear shifts one coordinate by an amount proportional to another coordinate. It changes the shape and orientation of a rectangular box, but it preserves its volume. To prove this without using a determinant formula, we compare box covers: a sheared box can be covered by axis-parallel boxes whose total volume is as close as desired to the original box’s volume.

Throughout, \(n\ge2\), and statements about measure refer to Lebesgue measure on the complete, sigma-finite measure space \((\mathbb{R}^n,\mathcal{L},m)\). Its outer measure is \(m^*\), defined by countable covers by axis-parallel boxes, with the volume of a box equal to the product of its side lengths. We use the outer-measure characterization of Lebesgue measurability: \(E\subseteq\mathbb{R}^n\) is in \(\mathcal{L}\) if and only if \(m^*(A)=m^*(A\cap E)+m^*(A\setminus E)\) for every \(A\subseteq\mathbb{R}^n\). All measure equalities below allow the value \(+\infty\).

The inverse formula follows directly: the \(j\)-th coordinate is unchanged, so subtracting \(t x_j\) from the new \(i\)-th coordinate recovers the original \(i\)-th coordinate. The case \(t=0\) is the identity map. We prove the volume statement for all \(t\), including this case.

Estimating the Image of a Box

Let \(Q=\prod_{k=1}^n(a_k,b_k)\) be a bounded open axis-parallel box with positive side lengths \(\ell_k=b_k-a_k\). Only the \(i\)-th coordinate changes under \(S_{ij,t}\). Subdivide the \(j\)-th interval into \(N\) equal pieces. On each resulting smaller box, the range of the new \(i\)-th coordinate has length at most \(\ell_i+|t|\ell_j/N\): its original range has length \(\ell_i\), and the change \(t x_j\) varies by at most \(|t|\ell_j/N\) on that piece. The \(j\)-th coordinate range there has length \(\ell_j/N\); all other coordinate lengths remain \(\ell_k\).

Consequently, the image of each smaller box is contained in an axis-parallel box of volume arbitrarily close to \((\ell_i+|t|\ell_j/N)(\ell_j/N)\prod_{k\ne i,j}\ell_k\). If the outer-measure definition uses open boxes, enlarge each containing box slightly; the total added volume can be made arbitrarily small because there are only finitely many pieces. Adding the \(N\) bounds and letting the enlargement tend to zero gives

As \(N\to\infty\), the second term tends to zero, so \(m^*(S_{ij,t}(Q))\le\prod_{k=1}^n\ell_k=m(Q)\). This estimate does not assume that the image is a box; it uses a finite cover of the image by axis-parallel boxes.

Proof. First suppose \(m^*(A)<\infty\), and let \(\varepsilon>0\). By the definition of outer measure, there is a countable cover of \(A\) by bounded axis-parallel boxes \(Q_r\) such that \(\sum_{r=1}^{\infty}m(Q_r)<m^*(A)+\varepsilon\). The box estimate gives \(m^*(S_{ij,t}(Q_r))\le m(Q_r)\) for every \(r\). Since \(S_{ij,t}(A)\subseteq\bigcup_r S_{ij,t}(Q_r)\), countable subadditivity of outer measure yields \(m^*(S_{ij,t}(A))\le\sum_r m^*(S_{ij,t}(Q_r))\le\sum_r m(Q_r)<m^*(A)+\varepsilon\). Letting \(\varepsilon\downarrow0\) proves \(m^*(S_{ij,t}(A))\le m^*(A)\). If \(m^*(A)=\infty\), this inequality holds automatically in the extended order. Apply the same inequality to \(S_{ij,-t}\) and the set \(S_{ij,t}(A)\). Since \(S_{ij,-t}(S_{ij,t}(A))=A\), it gives \(m^*(A)\le m^*(S_{ij,t}(A))\). Thus the outer measures are equal for every subset \(A\), including sets of infinite outer measure.

Now let \(E\in\mathcal{L}\). To check that \(S_{ij,t}(E)\) is measurable, take any \(A\subseteq\mathbb{R}^n\). The set \(S_{ij,-t}(A)\) is a subset of \(\mathbb{R}^n\), so measurability of \(E\) gives \(m^*(S_{ij,-t}(A))=m^*(S_{ij,-t}(A)\cap E)+m^*(S_{ij,-t}(A)\setminus E)\). The shear is a bijection, so the two sets on the right are respectively \(S_{ij,-t}(A\cap S_{ij,t}(E))\) and \(S_{ij,-t}(A\setminus S_{ij,t}(E))\). Outer-measure invariance, already proved for every subset, turns this equality into \(m^*(A)=m^*(A\cap S_{ij,t}(E))+m^*(A\setminus S_{ij,t}(E))\). The outer-measure criterion shows \(S_{ij,t}(E)\in\mathcal{L}\). Its measure equals its outer measure, which equals \(m^*(E)=m(E)\). \(\square\)

Rectangles Under a Shear

More generally, for a rectangle \([a_i,b_i]\times[a_j,b_j]\), its corner coordinates \((x_i,x_j)\) become \((x_i+t x_j,x_j)\). When \(t\ge0\), the new \(i\)-coordinate ranges from \(a_i+t a_j\) to \(b_i+t b_j\); when \(t<0\), it ranges from \(a_i+t b_j\) to \(b_i+t a_j\). The coordinate bounding box records the separate coordinate ranges, but need not equal the sheared rectangle. For a box in \(\mathbb{R}^n\), each vertex is transformed by the same coordinate rule, giving the vertices of its sheared parallelepiped.

Successive Shears and Parallelepipeds

A composition of shears also preserves Lebesgue measure: apply the theorem to the first image, which is measurable, and then apply it to the next image. The resulting set may be a parallelepiped whose edges are no longer aligned with the coordinate axes. Its measure is determined here by preservation at each step, not by a general determinant formula.

A Common Volume Mistake

A frequent error is to find the coordinate ranges of a sheared set, multiply their lengths, and call the result the volume of the set. For the planar example, that calculation gives \((9-2)(3-1)=7\cdot2=14\), but the sheared rectangle has measure \(6\). The calculation \(14\) is correct for the axis-parallel bounding box, not for the parallelogram inside it. The box-cover proof avoids this mistake: it approximates the sheared set itself with covers of controlled total volume, rather than replacing the set by its bounding box.

Elementary shears are important because they isolate a basic geometric operation: shifting one coordinate by another changes the orientation of a box without changing its Lebesgue measure. Together with translations, coordinate permutations, reflections, and diagonal scalings, they provide concrete tools for understanding linear transformations. The general volume rule for arbitrary linear maps requires an additional argument beyond the shear calculation.