Mr. Integral
Math notes/Linear Algebra/Linear Transformations

Linear Maps and Geometry — Cheat sheet

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Core rule

T(au+bv)=aT(u)+bT(v),T(a u+b v)=aT(u)+bT(v), A=[T(e1) ⋯ T(en)].A=[T(e_1)\ \cdots\ T(e_n)].

Watch for

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