3.3 Linear Transformations
Set a water molecule in the plane of the page: oxygen at the origin, the two hydrogens at (0.76, -0.59) and (-0.76, -0.59) Å, the twofold symmetry axis running along y. Chemists say water “has C_2 symmetry”: rotate it half a turn about that axis and the result is indistinguishable from the start — the hydrogens trade places, the oxygen stays put, and no experiment can detect that anything happened. The sentence sounds qualitative, but it is secretly a computation about every point of the molecule at once: a rule that sends each position (x, y) to a new position, applied to all atoms simultaneously, whose output configuration coincides with the input. Rules that move all of space — rotations, reflections, scalings — are called transformations, and this section shows that the well-behaved ones are precisely the matrices of §3.1 in their true vocation. A matrix is not a data table that happens to multiply; it is a motion of space, tabulated.
What makes a transformation linear
A transformation T assigns to every vector \mathbf{v} an output T(\mathbf{v}). The ones matrices can represent share two structural habits, and they are exactly the habits of the matrix–vector product itself (Exercise 7 of §3.1): they respect addition and scaling.
Definition 3.3.1 (Linear transformation). T is linear if for all vectors and scalars T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}), \qquad T(c\,\mathbf{v}) = c\, T(\mathbf{v}). Consequences: T(\mathbf{0}) = \mathbf{0} (take c = 0), and straight lines map to straight lines with even spacing preserved.
Rotation about the origin is linear — rotate two arrows and their tip-to-tail sum rotates along with them, since the whole parallelogram of §2.1 turns rigidly. So are reflections through lines containing the origin, and uniform stretches. The instructive failure is translation, “shift everything 2 Å along x”: it moves the origin, violating T(\mathbf{0}) = \mathbf{0}, so no matrix will ever represent it. (Crystallography needs translations badly and handles them by a bolt-on — affine transformations — but the symmetry operations of a molecule about its own center are exactly the linear kind, which is why matrices run point-group theory.)
The whole transformation in two columns
Linearity looks like a modest bookkeeping constraint. It is actually a devastating economy: a linear transformation of the plane is completely pinned down by what it does to just two vectors.
Theorem 3.3.2 (The matrix of a transformation). Let T be linear on the plane. Form the matrix M whose first column is T(\hat{\mathbf{x}}) and whose second column is T(\hat{\mathbf{y}}). Then T(\mathbf{v}) = M\mathbf{v} for every vector \mathbf{v}.
Proof. Any \mathbf{v} = (v_x, v_y) decomposes on the standard basis (§2.2): \mathbf{v} = v_x \hat{\mathbf{x}} + v_y \hat{\mathbf{y}}. Apply T and let linearity distribute it: T(\mathbf{v}) = T(v_x \hat{\mathbf{x}}) + T(v_y \hat{\mathbf{y}}) = v_x\, T(\hat{\mathbf{x}}) + v_y\, T(\hat{\mathbf{y}}). The right side is a linear combination of M’s columns weighted by \mathbf{v}’s entries — which is M\mathbf{v} by the column picture, Theorem 3.1.3. \square
The theorem cuts both ways, and both directions are daily tools. Building a matrix: ask where the basis vectors land, write the answers as columns, done. Reading a matrix: its columns are the fates of \hat{\mathbf{x}} and \hat{\mathbf{y}} — stare at the columns and the motion reveals itself. Build the two workhorses now. A rotation by angle \theta carries \hat{\mathbf{x}} to (\cos\theta, \sin\theta) — that is the definition of sine and cosine on the unit circle — and carries \hat{\mathbf{y}}, which starts 90° ahead, to (-\sin\theta, \cos\theta). Reflection across the y-axis flips \hat{\mathbf{x}} to -\hat{\mathbf{x}} and fixes \hat{\mathbf{y}}. Columns in place: R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}, \qquad \sigma_y = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}.
Before applying these to molecules, watch what a matrix does to space itself — not one vector but all of them at once. In the animation below, the entire coordinate grid glides from “nothing has happened” to the full action of the chosen matrix; the basis vectors \hat{\imath} and \hat{\jmath} ride the matrix’s columns (Theorem 3.3.2 made them the whole story), and the shaded unit square shows what becomes of area and orientation.
Example 3.3.3 (Water’s C_2, executed). With the axis along y, the C_2 operation is \sigma_y’s cousin: rotation by 180° about the y-axis in 3D, which acts on in-plane points as (x, y) \mapsto (-x, y) — the matrix \sigma_y above. Apply it to water’s three atoms and verify the symmetry claim.
Setting it up. Multiply each atomic position by the matrix; compare the output set of positions with the input set (atoms of the same element are interchangeable — a §1.2 sentiment).
Solution. \sigma_y \begin{pmatrix} 0.76 \\ -0.59 \end{pmatrix} = \begin{pmatrix} -0.76 \\ -0.59 \end{pmatrix}, \qquad \sigma_y \begin{pmatrix} -0.76 \\ -0.59 \end{pmatrix} = \begin{pmatrix} 0.76 \\ -0.59 \end{pmatrix}, \qquad \sigma_y \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}. H₁ lands exactly on H₂’s site, H₂ on H₁’s, O stays fixed: the transformed molecule occupies precisely the original sites.
Check. Lengths must survive a symmetry operation: each O–H distance is \sqrt{0.76^2 + 0.59^2} = 0.96 Å before and after ✓. The atom-swap is not an approximation — the coordinates were chosen symmetric, and the operation exchanges them exactly, which is what “the molecule possesses this symmetry” means. Run the same matrix on a hypothetical HOD molecule (one H heavier) and the output no longer matches the input — isotopic substitution genuinely lowers the symmetry, and the arithmetic shows it.
Composition, and the promise kept
Section 3.2 asserted that operations performed in different orders end differently, and promised a molecule to prove it. The matrix product was engineered so that applying B then A is the single transformation AB — composition is multiplication. So take a 90° rotation R = R(90°) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} and the reflection \sigma_x = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} (across the x-axis), and compose both ways: \sigma_x R = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}, \qquad R\,\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. Both products are reflections — but across different mirror lines (y = -x and y = x respectively; read each matrix’s columns to see where \hat{\mathbf{x}} and \hat{\mathbf{y}} go). Track a concrete atom at (1, 0): rotate-then-reflect sends it $ (1,0) (0,1) (0,-1)$, while reflect-then-rotate sends it (1,0) \to (1,0) \to (0,1). Different final positions, non-commutativity made visible — and this is not a pathology but the arithmetic backbone of group theory, where the multiplication table of a molecule’s symmetry operations is a fingerprint of its structure.
Transformation is re-arrangement
Step back far enough and this section is Section 1.3 again, grown up. Back there, the ARRANGEMENT primitive dealt in discrete slots: a permutation took one ordering of n objects and delivered another. Watch that idea put on matrix dress. The matrix P = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, applied to any vector, swaps its two components — read its columns: \hat{\mathbf{x}} goes to where \hat{\mathbf{y}} was and vice versa. It does not rotate, stretch, or blend; it re-seats. Matrices with exactly one 1 in each row and column — permutation matrices — are precisely the permutations of §1.3 acting on coordinate slots, and composing two of them multiplies the matrices, so the whole multiplication apparatus of §3.2 was secretly a theory of rearrangements all along. Water’s C_2 in Example 3.3.3 made the connection physical: the operation’s net effect on the atoms was a permutation — hydrogen 1 to hydrogen 2’s seat and back — which is exactly why the transformed molecule occupied the original sites.
The general linear transformation extends this from slots to the continuum: it re-arranges space itself, assigning every point a new address, all points simultaneously and coherently. One arrangement of the plane in, another out. But the extension comes with caveats, and the caveats are not fine print — they are the section’s main theorems seen from a new angle. First, not every conceivable rearrangement qualifies: a matrix can only perform rearrangements that fix the origin and keep lines straight with even spacing (Definition 3.3.1). Shifting everything sideways is a rearrangement, but not a linear one; bending the plane is a rearrangement, but no matrix bends. Second, and deeper: an honest rearrangement loses nothing — every output position is claimed by exactly one input, so the shuffle can be undone. Invertible matrices are honest in exactly this sense. A singular matrix is not: the projection P of Exercise 9 sends (1, 0) and (1, 5) to the same seat, and once two occupants share a chair, no rule can send each back where it came from. A singular matrix is therefore not a rearrangement of space but a forgetting of it — which is §3.2’s “no inverse exists,” re-derived from the primitive. Third, among the honest rearrangements sits the aristocracy that preserves the arrangement’s own internal geometry — every distance, every angle — and those, the orthogonal matrices of Exercise 8, are precisely the operations a molecule can call its symmetries.
So the ARRANGEMENT primitive has two faces, and Chapter 3 has now shown both. Its static face is §3.1: a grid records an arrangement — which entry sits in which row and column is the information. Its dynamic face is this section: a matrix carries one arrangement to another. They are not two applications of matrices but one idea seen at rest and in motion, and the next section’s determinant is the natural auditor of the motion: one number reporting whether the rearrangement preserved the arrangement’s volume, whether it flipped its handedness, and — when the number is zero — whether it was ever a rearrangement at all.
Remarks and cautions
Read matrices by their columns. Handed an unfamiliar 2\times2, resist entry-by-entry staring: column 1 is where \hat{\mathbf{x}} goes, column 2 is where \hat{\mathbf{y}} goes, and Theorem 3.3.2 says that is the whole story. \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} swaps the basis vectors — it can only be the reflection across y = x.
Symmetry operations are the length-preserving minority. Every symmetry operation is linear, but most matrices are not symmetry operations: the shear and stretch in the widget mangle bond lengths, which no physical symmetry may do. The matrices that preserve all lengths and angles have orthonormal columns (Exercise 8) — the orthogonal matrices, and molecular point groups live entirely inside that aristocratic subset.
Translation is not linear. A matrix fixes the origin, always. Moving a molecule bodily across the cell is a different mathematical object (affine), which is why symmetry analysis pins the point group at the molecule’s center — the one point every operation leaves alone.
Reflections reverse handedness. Apply \sigma_y to your right hand and the output is a left hand; no rotation can undo it. Two numbers in the matrix know this — the next section shows that a single computed quantity distinguishes proper rotations from improper reflections, and its sign is exactly the handedness flag chirality has been waiting for.
Summary
A linear transformation is a motion of all space that respects vector addition and scaling — rotations, reflections, and stretches about the origin, but never translations — and Theorem 3.3.2 compresses any such motion into a matrix whose columns are simply the destinations of the basis vectors, making matrices buildable (write the columns) and readable (read them) in one stroke. Rotation and reflection matrices follow instantly, water’s C_2 becomes an exact arithmetic identity on its coordinates, and composition of motions is the matrix product of §3.2, whose non-commutativity now has geometric flesh: rotate-then-reflect and reflect-then-rotate are different mirrors. Seen from the primitives, the section closes a circle begun in §1.3: a transformation is one arrangement carried to another — permutation matrices doing it to discrete slots, general matrices doing it to space itself — with the caveats doing the real work: only origin-fixing, line-preserving rearrangements are linear, only invertible ones lose nothing, and the length-preserving orthogonal minority are the physical symmetry operations. One question now stands open with everything staged for it: what single number audits the rearrangement — its effect on volume, on handedness, and on whether it was a rearrangement at all? That number is the determinant, next.
Exercises
Write the matrices for: rotation by 270°; reflection across the line y = x; the stretch that doubles x and leaves y alone; and rotation by 120° (exact entries, using \cos 120° = -\tfrac12, \sin 120° = \tfrac{\sqrt3}{2}).
Identify each matrix from its columns, in words: \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}; \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}; \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}; \begin{pmatrix} 1 & 0.4 \\ 0 & 1 \end{pmatrix}.
Ammonia’s C_3 axis: apply R(120°) (Exercise 1) three times in succession to the point (1, 0) and verify it returns home — algebraically, R(120°)^3 = I. Why must every rotation matrix of a molecular symmetry axis satisfy R^n = I for some n?
Verify by direct multiplication the two compositions of the text, \sigma_x R(90°) and R(90°)\,\sigma_x, and confirm the trajectories of the atom at (1, 0) given there.
Benzene in the plane: its carbons sit at angles 0°, 60°, \ldots, 300° on a circle of radius 1.39 Å. Show that R(60°) maps the carbon set to itself, and find the smallest positive rotation that does the same for the hydrogens of a hypothetical C₆H₃ (alternating substitution at carbons 1, 3, 5).
Which of these are linear transformations? Justify with the two defining habits or a violated consequence: (a) T(x, y) = (2x - y, x); (b) T(x, y) = (x + 1, y); (c) T(x, y) = (x^2, y); (d) $T() = $ projection of \mathbf{v} onto a fixed line through the origin (§2.3’s \operatorname{proj}).
★ Prove that rotation about the origin is linear, geometrically: argue that rotating a parallelogram rigidly rotates its diagonal, so R(\mathbf{u} + \mathbf{v}) = R\mathbf{u} + R\mathbf{v}, and similarly for scaling. (No components needed — that is the point.)
★ Show that a matrix preserves all lengths if and only if its columns are orthonormal. Forward direction: lengths of M\hat{\mathbf{x}}, M\hat{\mathbf{y}}, and M(\hat{\mathbf{x}} + \hat{\mathbf{y}}) pin down the columns’ norms and mutual dot product. Confirm that R(\theta) passes for every \theta and the shear fails.
★ The projection matrix onto the x-axis is P = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}. Compute P^2, explain the answer in words (project twice — then what?), and explain why no inverse of P can exist, first physically (what information is destroyed?) and then via §3.2’s cancellation failure: find \mathbf{u} \neq \mathbf{v} with P\mathbf{u} = P\mathbf{v}.
★★ Chirality by arithmetic: take the “right-handed” tripod of points (1,0), (0,1) and their sum (1,1), apply \sigma_y, and show that no rotation R(\theta) maps the reflected tripod back onto the original (compare, for the images of (1, 0) and (0, 1), the sense — clockwise or counterclockwise — in which one must turn to reach the other). Every chiral molecule is this exercise wearing atoms; the sign that automates the check arrives in §3.4.
★ Permutations in matrix dress: (a) verify that P = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} swaps the components of any vector (x, y); (b) write the 3 \times 3 matrix performing the cyclic re-arrangement (x, y, z) \mapsto (z, x, y), and check it on (1, 2, 3); (c) how many 3 \times 3 permutation matrices exist? (§1.3 already answered this — say which formula); (d) compute \det P for the swap in (a), and explain what its sign says about a single swap, anticipating §3.4.