Linear Transformations Summary

Linear Algebra (Matrices) · Axiom Academy

SUMMARY Unit Summary: Linear Transformations Functions between vector spaces that preserve structure — and how every one of them is, at heart, a matrix. A map is linear exactly when it respects addition and scaling: T(u+v)=T(u)+T(v) and T(cu)=cT(u) . Every linear is a matrix , and T(x)=Ax . The kernel and image read off injectivity and surjectivity; rank–nullity ties their sizes together: . Composition is matrix product , inverse is matrix inverse, and A is invertible T is bijective . Changing basis conjugates the matrix: B=P^ -1 AP (similar matrices share rank, det, trace, eigenvalues); diagonalization is just the basis of eigenvectors. Core Concept What Makes a Map Linear A linear transformation preserves the two vector-space operations. Both conditions fold into one test: T(au+bv)=aT(u)+bT(v) . Always: T(0)=0 , and lines through the origin map to lines (or a point). Watch out for: a constant shift like T(x)=x+1 is not linear — it moves the origin. Core Concept The Standard Matrix Apply T to each standard basis vector; the outputs become the columns of A . Computing T is then just matrix–vector multiplication. When to use: any — read its action one basis vector at a time. Watch out for: the matrix depends on the chosen bases; different bases give different matrices for the same T . Core Concept Geometric Transformations

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