ALGORITHMS

• Systems:
• Consistency
• General solution
• Vector spaces:
• Basis for:
• row space
• column space
• orthogonal complement
• span of a set of vectors
• eigenspace
• Equations defining a subspace (given a spanning set)
• Basis for the intersection of two subspaces (given spanning sets for each).
• Orthonormal basis (given a basis)
Matrices:
• rank and nullity
• LU decomposition
• Inverse of a matrix
• determinant
• eigenvalues
• characteristic polynomial
• diagonalization
• powers
• orthogonal projection
• orthogonal diagonalization
• spectral decomposition

APPLICATIONS:

• Polynomial interpolation
• Markov processes
• Cramer's rule
• Differential equations
• Moore graphs
• Least squares
• Conic sections

CONCEPTS

• Matrices
• matrix
• symmetric matrix
• transpose
• rank
• row echelon form
• reduced row echelon form
• matrix-vector product
• linear transformation
• identity matrix
• rotation
• reflection
• matrix product
• inverse
• invertibility
• elementary matrix
• determinant
• characteristic polynomial
• eigenvalue
• eigenvector
• diagonalization
• similar matrices
• orthogonal matrix
• Vectors
• vector
• linear combination
• span
• subspace
• linear independence
• basis
• dimension
• norm
• inner product
• orthogonality
• orthogonal projection
• orthogonal complement

THEOREMS

• A linear system has either 0, 1, or infinitely many solutions.
• A system of linear equations is solvable if and only if the data column of the augmented matrix lies in the span of the coefficient columns.
• Row rank equals column rank; a matrix and its transpose have the same rank.
• Any two matrices related by a series of elementary row operations have the same row space.
• Any two matrices related by a series of elementary row operations have the same nullspace.
• Any linear relation among the columns of the reduced row echelon form of a matrix is valid for the columns of the original matrix.
• The columns in a matrix containing the pivot positions for reduced echelon form give a basis for the column space.
• The nonzero rows of any row echelon form of a matrix give a basis for its row space.
• The ``general solution'' of a homogeneous system gives a basis for the null space of the coefficient matrix.
• The reduced row echelon form of a matrix is unique.
• Matrix-vector multiplication preserves the vector operations.
• Elementary row operations are performed by multiplication by elementary matrices on the left; multiplication on the right performs the corresponding elementary column operations.
• Every invertible matrix is a product of elementary matrices
• A product of invertible matrices is invertible
• The transpose of an invertible matrix is invertible.
• Determinant test for invertibility
• The determinant of a product is the product of the determinants.
• Effect of elementary row operations on determinants.
• The eigenvalues are the roots of the characteristic polynomial.
• The geometric multiplicity of an eigenvalue is no greater than the algebraic multiplicity.
• Similar matrices have the same determinant, the same characteristic polynomial, and the same eigenvalues - with the same multiplicities (both algebraic and geometric).
• If two matrices are similar to a third, they are similar to each other.
• The span of a set of vectors gives a subspace of R^n.
• The orthogonal complement of a set of vectors gives a subspace of R^n.
• The row space, column space, and null space of a matrix are subspaces of R^n.
• The rank of a matrix is equal to the dimension of its row space as well as its column space.
• The nullity of a matrix is equal to the dimension of its null space.
• The rank plus the nullity of a matrix add up to its width.
• If W is a subspace of R^n of dimension d, then the dimension of its orthogonal complement is n-d.
• The column space of a matrix A is the image of the function f(x)=Ax.
• Any two bases for a subspace have the same number of elements
• A linearly independent set never has more elements than a spanning set
• A homogeneous system with more variables than equations has a nontrivial solution
• Any set of more than n vectors in R^n is linearly dependent.
• If a set of vectors is linearly dependent, then one of them is in the span of the others.
• Orthogonal, nonzero vectors are linearly independent
• Eigenvectors corresponding to distinct eigenvalues of a matrix are linearly independent; and if the matrix is symmetric, they are orthogonal.
• The transpose of an orthogonal matrix is orthogonal
• A product of two orthogonal matrices is orthogonal.
• A matrix is orthogonal if and only if multiplication by that matrix preserves lengths of vectors; in this case, angles are also preserved.
• For any matrix A, the null space of A^TA and of A are the same.
• If the columns of a matrix A are linearly independent, then A^TA is invertible.
• Au.v=u.A^Tv (and A^T is the only matrix related to A in this way).
• Orthogonal projection matrices are symmetric, and satisfy P^2=P.
• If a symmetric matrix P satisfies P^2=P, then P is the orthogonal projection matrix onto its own column space, and I-P is the orthogonal projection matrix onto its null space.
• If A is a matrix with linearly independent columns, then the orthogonal projection onto its column space is given by the matrix A(A^TA)^-1 A^T. If an nxn matrix has n distinct real eigenvalues, then it is diagonalizable.
• An nxn matrix is diagonalizable if and only if its eigenvalues are real, and each one has algebraic multiplicity equal to its geometric multiplicity.
• Cauchy-Schwartz
• Triangle inequality
• For any vector u and any subspace W, the closest vector in W to u is the orthogonal projection of u to W, and the distance from u to W is the length of their difference.
• For any subpace W and any vector u, there is a unique decomposition of u as the sum of two vectors, one in W and one orthogonal to W.

MORE ADVANCED RESULTS

• All of the eigenvalues of a symmetric matrix are real; and the matrix is diagonalizable by an orthogonal matrix. Any nxn matrix has n (complex) eigenvalues, counting multiplicities.
• Any function which preserves the vector operations (scalar multiplication and vector addition) is given by matrix multiplication; that matrix is unique.
• Two matrices are similar if they represent the same linear transformation, with respect to different bases.