Linear Transformations
Cheat sheet.

The key rules, formulas and reminders from all 3 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Linear Maps and Geometry

T(a u+b v)=aT(u)+bT(v),

A=[T(e_1)\ \cdots\ T(e_n)].

A map taking zero to zero need not be linear. One-to-one refers to the kernel; onto refers to the stated codomain.

x=P[x]_B,\quad [x]_B=P^{-1}x,\quad A_B=P^{-1}AP.

Read matrix products from right to left. A basis change changes coordinates, not the underlying vector or operator.

$\det(AB)=\det A\det B$ . Row swap: negate. Row scale: scale. Add a row multiple: unchanged. Triangular determinant: diagonal product.

Determinants apply to square matrices. Magnitude measures volume scaling, not by itself numerical conditioning.

2 reference blocks

T(a u+b v)=aT(u)+bT(v),

A=[T(e_1)\ \cdots\ T(e_n)].

A map taking zero to zero need not be linear. One-to-one refers to the kernel; onto refers to the stated codomain.

2 reference blocks

x=P[x]_B,\quad [x]_B=P^{-1}x,\quad A_B=P^{-1}AP.

Read matrix products from right to left. A basis change changes coordinates, not the underlying vector or operator.

2 reference blocks

$\det(AB)=\det A\det B$ . Row swap: negate. Row scale: scale. Add a row multiple: unchanged. Triangular determinant: diagonal product.

Determinants apply to square matrices. Magnitude measures volume scaling, not by itself numerical conditioning.