Transposition flips a matrix across its top-left-to-bottom-right diagonal: rows become columns, and columns become rows. This simple rearrangement often explains why matrix code appears to use the wrong multiplication order. section 2.5.2
Use transposition to make matrix conventions explicit:
- Check whether a library stores vectors as rows or columns before combining its transformation matrices with another library. section 4.2.1
- Treat repeated transpositions inside one routine as a warning sign: the code may have been adjusted by trial and error rather than designed consistently. section 2.5.2
- Choose one vector-and-matrix convention wherever possible so transformations remain readable and easier to reason about. section 4.2.1
A concrete transpose
Starting with [[1, 2, 3], [4, 5, 6], [7, 8, 9]], transpose the rows into columns to obtain [[1, 4, 7], [2, 5, 8], [3, 6, 9]]. The operation changes layout, not the underlying entries. section 2.5.2
[ [1, 2, 3], [4, 5, 6], [7, 8, 9] ]
Inspect the matrix listing used to visualize the entries before exchanging its rows and columns. section 2.5.2
