1. Overview¶
Linear algebra provides a convenient and general way to pose and analyze problems in multiple dimensions. The most tractable and universal such problems involve linear transformations.
A function \(L\) on domain \(D\) is linear if it satisfies both of these rules:
and
I’ve been deliberately vague about the domain and range of \(L\) in this definition, because it can be adapted to multiple contexts.
1.1. Glossary¶
- algebraic multiplicity¶
For an eigenvalue, its multiplicity in the sense of its appearance in the factorization of the characteristic polynomial.
- augmented matrix¶
Result of horizontally concatenating the coefficient matrix of a linear system with the right-side vector.
- basis¶
Minimal set of constant vectors needed to describe the general solution of a homogeneous linear system.
- charcteristic polynomial¶
Polynomial of degree \(n\) whose roots are the eigenvalues of an \(n\times n\) matrix.
- coefficient matrix¶
Matrix of all coefficients multiplying unknowns in a linear algebraic system of equations.
- cofactor expansion¶
Definition and/or computational method for finding the determinant of a matrix through recursive size reduction.
- conjugate¶
Replacement of each occurence of \(i\) by \(-i\) in any representation of a complex number, i.e., reflection about the real axis in the complex plane.
- Cramer’s Rule¶
Algorithm to compute the solution of a linear algebraic system using only determinant calculations.
- defective¶
For an eigenvalue, having geometric multiplicity less than the algebraic multiplicity; for a matrix, having one or more defective eigenvalues.
- determinant¶
Scalar value computed from a square matrix that is zero if and only if the matrix is singular.
- eigenspace¶
General solution of \((\bfA-\lambda\meye)\bfv=\bfzero\) for a given square matrix \(\bfA\) and some nonzero vector \(\bfv\).
- eigenvalue¶
Scalar value \(\lambda\) satsifying \(\bfA\bfv=\lambda\bfv\) for a given square matrix \(\bfA\) and some nonzero vector \(\bfv\).
- eigenvector¶
Nonzero vector \(\bfv\) satsifying \(\bfA\bfv=\lambda\bfv\) for a given square matrix \(\bfA\) and some scalar \(\lambda\).
- free column¶
Column of a RREF matrix that contains no leading ones.
- Gaussian elimination¶
Use of row operations to transform an augmented matrix to a triangular form.
- geometric multiplicity¶
For an eigenvalue, the number of basis vectors in its associated eigenspace.
- general solution¶
Representation of all possible solutions to a linear algebraic system of equations.
- homogeneous¶
Linear system of equations with zero right side, i.e., \(\bfA\bfx=\bfzero\).
- identity matrix¶
Square diagonal matrix with ones on the diagonal, serving as the multiplicative identity.
- imaginary part¶
Component of a complex number that multiplies the imaginary unit \(i\) when the number is written in standard rectangular form.
- inconsistent¶
Linear algebraic system of equations that has no solution.
- invertible¶
Matrix that has an inverse and which gives a unique solution for every linear algebraic system.
- inverse¶
Unique matrix that multiplies a given square invertible matrix to produce the identity matrix.
- leading nonzero¶
First (i.e., leftmost) nonzero element of a matrix row.
- linear¶
Type of function or operator that satisfies two basic properties.
- linear combination¶
Simultaneous multiplication of vectors by coefficients, followed by summation to a single vector.
- matrix¶
Array of numbers obeying certain algebraic rules.
- modulus¶
Extension of absolute value to complex numbers, representing the distance between values in the complex plane.
- particular solution¶
Any one solution of linear algebraic system, as opposed to the general solution.
- pivot column¶
Column of an RREF matrix that contains a leading one.
- real part¶
Component of a complex number that does not multiply the imaginary unit \(i\) when the number is written in standard rectangular form.
- row elimination¶
Use of elementary operations on the rows of a matrix to transform it to a simpler one, such as its RREF equivalent.
- RREF¶
Matrix meeting certain requirements that expose all necessary information about solving a linear algebraic system; stands for “reduced row echelon form.”
- scalar¶
Real or complex number, as distinguished from a vector or matrix.
- scalar multiplication¶
Multiplication of each element of a vector or matrix by a given number.
- singular¶
Matrix that does not have an inverse and for which homogeneous linear equations have infinitely many solutions.
- square¶
Matrix with the same number of rows as columns.
- vector¶
Finite collection of numbers obeying certain algebraic rules.