Steply

Linear Algebra formula sheet

14 formulas · what each one means · a free lesson for each

All formula sheets

steplyclass.com/formula-sheets/linear-algebra/ · free, step-by-step lessons for every formula

Systems and spaces

Linear combination
c1v1 + c2v2 + … + cpvp
Lesson: Linear combinations →
Matrix equation
Ax = b
Row reduce [A | b] to find the solution set.Lesson: Solution sets →
Linear independence
c1v1 + … + cpvp = 0 only when all c = 0
Lesson: Linear independence →
Null space
Nul A = {x : Ax = 0}
Lesson: Null space →
Rank theorem
rank A + dim Nul A = n
n = number of columns.Lesson: Rank and column space →
Matrix transformation
T(x) = Ax
The columns of A are where the basis vectors go.Lesson: Matrix transformations →

Determinants and eigenvalues

Determinant rules
det(AB) = det A × det B det AT = det A
Swap two rows: × −1. Multiply a row by k: × k.Lesson: Determinant rules →
Characteristic equation
det(A − λI) = 0
Its roots are the eigenvalues λ.Lesson: Eigenvalues →
Eigenvectors
(A − λI)v = 0
Lesson: Eigenvectors →
Diagonalize
A = PDP−1
P = eigenvectors as columns, D = eigenvalues on the diagonal.Lesson: Diagonalizing a matrix →

Lengths and projections

Length
‖v‖ = √v • v
Lesson: Dot product and length →
Projection
proju y = y • uu • uu
Lesson: Projection onto a vector →
Gram-Schmidt
v2 = x2 − x2 • v1v1 • v1v1
Take away the part along the vectors already found.Lesson: Gram-Schmidt →
Least squares
AT Ax̂ = ATb
Lesson: Least squares line →

More formula sheets