Proof of the Matrix Representation Theorem for Linear Transformations

Theorem: Matrix Representation of Linear Transformations

Proof

We prove both directions of the theorem.

For the forward direction, we use the vector addition and scalar multiplication properties of matrices to show that T(v) = Av defines a linear transformation.

Handwritten mathematical proof of the matrix representation theorem for linear transformations, slide 4: proof of scalar multiplication.

Handwritten mathematical proof of the matrix representation theorem for linear transformations, slide 5: completing the forward implication.

Handwritten mathematical proof of the matrix representation theorem for linear transformations, slide 6: beginning the converse proof.

For the backwards direction, we assume T is linear and express an arbitrary vector as a linear combination of standard basis vectors to construct A from the columns T(e₁), T(e₂), …, T(eₙ).