Math 060 101817C Matrix Transformations of Linear Transformations

Recall: If linear transformations agree on a basis, then they are equal. Recall: To each matrix A corresponds a linear transformation (left-multiplication by A). Theorem: Every linear transformation between Euclidean spaces corresponds to left-multiplication by some matrix. Example. Exercise: find the matrix. Concept: matrix representation of a linear transformation between vector spaces with respect to (two) bases. Theorem: Every linear transformation between (abstract) vector spaces (given bases of those spaces) has a matrix representation with respect to those bases (proof deferred to next time). Example: differentiation on the polynomials of degree less than three.