Three-dimensional linear transformations | Chapter 5, Essence of linear algebra

Linear transformations in three dimensions are fully determined by where they send the three standard basis vectors—so a 3D “grid-squishing” process can be encoded by just nine numbers in a 3×3 matrix. The key move is to imagine every point in space as the tip of a vector, then require that the transformation keeps grid lines parallel and evenly spaced while fixing the origin. Under those rules, the entire mapping from input vectors to output vectors is captured once the images of i-hat, j-hat, and k-hat are known.

In 3D, the standard basis vectors are the unit directions along the x, y, and z axes: i-hat, j-hat, and k-hat. Instead of tracking the full 3D grid (which quickly becomes visually cluttered), the transformation can be understood by following only these three axes. Record the coordinates of where i-hat lands, where j-hat lands, and where k-hat lands; then place those three coordinate triples as the columns of a 3×3 matrix. That matrix becomes a complete description of the linear transformation.

A concrete example makes the encoding feel tangible: a 90-degree rotation around the y-axis. Under this rotation, i-hat moves to the point (0, 0, −1), j-hat stays put at (0, 1, 0), and k-hat moves to (1, 0, 0). Using those three column vectors, the resulting 3×3 matrix stores the rotation in a compact form.

To apply the transformation to an arbitrary vector with coordinates (x, y, z), the process mirrors the 2D case. Each coordinate acts like an instruction for how much to scale the corresponding basis vector, and the scaled basis vectors add up to form the original vector. Crucially, this “scale-and-add” structure is compatible with the transformation: the same linear combination logic works before and after applying the matrix. Practically, that means multiplying the vector’s coordinates against the matrix’s columns and summing the results to get the output coordinates.

Matrix multiplication then takes on a geometric meaning: when two 3×3 matrices are multiplied, the right-hand matrix’s transformation happens first, followed by the left-hand matrix’s transformation. This ordering matters because the product represents composition of linear maps, not just number crunching. That composition viewpoint is especially important in computer graphics and robotics, where rotations in 3D are easier to understand and implement when they’re broken into successive, simpler rotations.

Finally, the discussion points toward the next topic—determinants—signaling that the next step will be about extracting deeper information from matrices beyond how they move basis vectors and compose transformations.