*This is a series of notes taken during my review of linear algebra, using Axler's excellent textbook Linear Algera Done Right, which will be heavily referenced.*

This is based on book section (3.C).

Suppose \(T \in \mathcal{L}(V, W)\) and \(v_1,...,v_n\) is a basis of \(V\) and \(w_1,...,w_m\) is a basis of \(W\). The matrix of \(T\) with respect to these bases is the \(m\)-by-\(n\) matrix \(\mathcal{M}(T)\) whose entries \(A_{j,k}\) are defined by

The \(k\)th column of \(\mathcal{M}(T)\) consists of the scalars needed to write \(T v_k\) as a linear combination of \(w_1,...,w_m\).

Example (3.33): Suppose \(T \in \mathcal{L}(\mathbb{F}^2,\mathbb{F}^3)\) is defined by

Find the matrix of \(T\) with respect to the standard bases of \(\mathbb{F}^2\) and \(\mathbb{F}^3\).

Solution: Because \(T(1, 0)=(1,2,7)\) and \(T(0,1)=(3,5,9)\), the matrix of \(T\) with respect to the standard bases is then

**Follow-up question**: find the matrix of \(T\) with respect to the basis \((1,1), (-1,1)\) of \(\mathbb{F}^2\) and \((1,3,0),(0,-1,0),(0,0,2)\) of \(\mathbb{F}^3\).

Solution: Because \(T(1,1) = (4,7,16) = 4(1,3,0) + 5(0,-1,0) + 8(0,0,2)\), and \(T(-1,1) = (2,3,2) = 2(1,3,0) -3(0,-1,0) + (0,0,2)\), the matrix with respect to these (non-standard) bases is

**Key point: the matrices can change with bases, but the underlying operator stays the same!** Later on we will see how trace and determinant are really properties of operators, not just their matrix representations.