58. Matrix Representation

Every linear transformation can be uniquely represented by an matrix such that:

Conversely, every matrix gives a linear transformation. So linear transformations are the same as matrices.

58.1. How to find the matrix

Apply to each standard basis vector . The results become the columns of :

58.2. Why this works (sketch)

Any input is a linear combination of standard basis vectors:

Applying and using linearity:

The columns of are exactly the images of the standard basis under .

58.3. Identity transformation

The identity , , has matrix (the identity matrix), since for each standard basis vector.

58.4. See also

• Linear Transformation — definition
• Matrix–Vector Product — the operation that makes a matrix into a transformation
• Change of Basis — what happens to the matrix when you switch bases
• Rotation Matrix, Reflection Matrix, Scaling Matrix, Shear Matrix — common examples