mat 223 theorems

Theorem 1.19: Gauss Jordan

Every matrix A is row equivalent to a matrix in row echelon form X. Furthermore, if X is in reduced row echelon form, then this matrix is unique.

Theorem 1.27: Rouché-Capelli

Suppose that a system of linear equations has augmented matrix A and coefficient matrix C. Then,

(1) The system is inconsistent if and only if the last column of rref(A) has a pivot.

(2) The system has exactly one solution if and only if the last column of rref(A) does not have a pivot, and every column of rref(C) has a pivot.

(3) The system has infinitely many solutions if and only if the last column of rref(A) does not have a pivot and rref(C) has a column without a pivot.

Theorem 4.4: Subspace-Span Theorem

A subset V is a subspace of R^n if and only if there exist vectors v1, v2, …, vm so that V = Span(v1, v2, …, vm).

Theorem 4.11: Finding Bases

Let V be the vector subspace of Rn given by

V = Span(v1, …, vm).

If A is the matrix with column vectors v1, …, vm then the pivot columns of A will form a basis for V . Furthermore, if the reduced row echelon form of A has k pivots, then dim(V ) = k.

Corollary 5.9: The Rank-Nullity Theorem

Let A be an m × n matrix. Then, rank(A) + nullity(A) = n.

Theorem 7.17: The Invertible Matrix Theorem

Let A be an n × n matrix. Then, the following are equivalent:

(1) A is invertible;

(2) The matrix-vector equation Ax = b has a unique solution for any b element R^n;

(3) rref(A) = In;

(4) A is a product of elementary matrices.

Theorem 11.13: The Diagonalization Theorem

Let A be an n × n matrix whose eigenvalues are all real. The following are equivalent.

(1) A is diagonalizable;

(2) The sum of the geometric multiplicities of A is equal to n;

(3) The geometric multiplicity of every eigenvalue lambda is equal to the algebraic multiplicity of lambda.

Theorem 12.8: The Gram-Schmidt Process

Every vector space has an orthogonal basis. Furthermore, if V is a vector subspace of Rn with basis {v1, v2, …, vm},

and we let

u1 = v1

u2 = v2 - proju1v2

u3 = v3 - proju1v3 - proju2v3

…

um = vm - proju1vm - projusvm - … - projum-1vm, then, {u1, …, um} is an orthogonal basis for V.