Finding the Matrix

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Finding the Matrix of a Linear Transformation Build the standard matrix of a transformation by computing its action on the basis vectors. The linear transformation rotates every vector counterclockwise, which algebraically is . Find its standard matrix A , then verify that on the test vector . Nice work! You built the matrix of a linear transformation straight from its action on the basis vectors, then verified it. Here is the recipe to keep: The columns rule: the standard matrix is — each column is where T sends a basis vector. Why it works: every is a combination of the basis vectors, and T is linear, so . The result: the rotation has matrix , and A(3,2) = (-2,3) = T(3,2) . The same recipe builds the matrix of any linear transformation — rotations, reflections, projections, or one given by a formula. Compute the images of the basis vectors and stack them as columns.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.