Matrix multiplication as composition | Chapter 4, Essence of linear algebra

Matrix multiplication isn’t just a computational trick—it’s a compact way to represent composing linear transformations. A linear transformation is determined entirely by where it sends the basis vectors i-hat and j-hat, because any vector (x, y) can be written as x·i-hat + y·j-hat. Once the transformed positions of i-hat and j-hat are known, the transformed landing spot of any vector follows automatically: x times the transformed i-hat plus y times the transformed j-hat. Recording those transformed basis-vector coordinates as the columns of a matrix turns that geometric rule into matrix-vector multiplication, which is the computational meaning of “apply the transformation.”

That framework becomes especially powerful when two transformations must be applied in sequence. If one matrix represents a rotation and another represents a shear, applying them one after the other produces a new linear transformation with its own matrix. The key construction is column-by-column: the first column of the composition matrix is what you get by sending i-hat through the right-hand matrix first, then through the left-hand matrix; the second column comes from the same process for j-hat. This directly matches the rule for matrix multiplication: multiplying two matrices corresponds to applying the right matrix first, then the left matrix. That right-to-left order can feel unintuitive at first, but it’s consistent with function composition—variables are written on the right, so the “later” transformation appears on the right.

A concrete example makes the mechanics clear. One matrix, M1, has columns (1, 1) and (-2, 0), and another, M2, has columns (0, 1) and (2, 0). To build the matrix for “apply M1 then M2,” the first column comes from applying M2 to the vector where i-hat lands after M1: M1 sends i-hat to (1, 1), and then M2 sends (1, 1) to (2, 1). The second column comes from applying M2 to where j-hat lands after M1: M1 sends j-hat to (-2, 0), and M2 sends that to (0, -2). The resulting composition matrix captures the combined effect as a single transformation.

Thinking in terms of transformations also resolves two classic algebraic questions about matrix multiplication. Order matters: composing a shear with a 90-degree rotation in one order produces a different final placement of i-hat and j-hat than composing them in the opposite order, so the product depends on which matrix comes first. Meanwhile, associativity holds: for three matrices A, B, and C, the placement of parentheses doesn’t change the outcome because the same sequence of transformations is applied either way—first C, then B, then A. Numerically proving associativity can be messy, but conceptually it’s immediate: both parenthesizations apply the identical chain of operations in the same order. The takeaway is that matrix multiplication’s algebraic properties reflect the geometry of composing linear maps, not just memorized arithmetic rules.