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This lecture shows how to find matrix representations of linear maps by matching coefficients, introduces special matrix types (diagonal, triangular, permutation), derives the matrix product formula from sequential application of linear maps, and defines matrix inverses. Lecturer: Dr Markus Brede.

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📋 At a Glance

• [ ] Find the matrix representation of a linear map by writing the map in coordinates and matching coefficients
• [ ] Recognise when a map cannot be represented by a matrix (e.g. normalisation — not linear)
• [ ] Identify special matrices: identity, diagonal, upper/lower triangular, block diagonal, permutation
• [ ] Construct permutation matrices that swap given vector components
• [ ] Understand sequential application of maps: $x \xrightarrow{A} y \xrightarrow{B} z = (BA)x$
• [ ] Derive and use the matrix product formula: $C_{ij} = \sum_k B_{ik} A_{kj}$ (row $i$ of $B$ × column $j$ of $A$)
• [ ] Verify dimension compatibility: $(m \times p) \cdot (p \times n) = (m \times n)$
• [ ] Know the properties of matrix multiplication: distributive ✓, associative ✓, commutative ✗
• [ ] Construct a counterexample to show $AB \neq BA$
• [ ] Know the effect of left-multiplying vs right-multiplying by a permutation matrix (swap rows vs columns)
• [ ] State the definition of a matrix inverse: $AB = BA = I_n$
• [ ] Verify whether a given matrix is the inverse of another by checking both products

1. Finding Matrix Representations of Linear Maps

The Method: Coefficient Matching

Given a linear map described geometrically, find its matrix by:

1. Write the map in coordinates: what happens to each component?