Geometry and algebra of linear maps, vectors, and matrices
The transpose Aᵀ flips a matrix across its main diagonal: rows become columns and columns become rows. Entry (i, j) swaps with entry (j, i). An (m×n) matrix becomes (n×m).
Imagine a spreadsheet where rows are people and columns are the months they each paid. Transposing it tips the whole table on its diagonal so rows become columns: now rows are months and columns are people. No number is lost or changed — each value just moves to its mirrored cell, where its row label and column label have traded places.
A matrix that equals its own transpose, A = Aᵀ, is symmetric: mirror-balanced across the diagonal, with Aᵢⱼ = Aⱼᵢ. These matrices are special enough that two whole later lessons are devoted to them.