MATH 2320 01 - Linear Algebra Vocab

Linear Equation

An equation that can be written in the form a1x1+a2x2+...+anxn=b, where b and the coefficients a1,...,an are real or complex numbers.

System of linear equations

A collection of one or more linear equations involving the same set of variables, say, x1,...,xn.

leading entry

The leftmost nonzero entry in a row of a matrix.

pivot position

A position in a matrix A that corresponds to a leading entry in an echelon form of A.

basic variable

A variable in a linear system that corresponds to a pivot column in the coefficient matrix.

linear combination

A sum of scalar multiples of vectors. The scalars are called the weights.

Product Ax

The linear combination of the columns of A using the corresponding entries in vector x as weights.

Homogeneous equation

An equation of the form A(vector)x=(vector)0, possibly written as a vector equation or as a system of linear equations.

linearly independent vectors

An indexed set {vector v1,…,vector vp} with the property that the vector equation c1 vector v1+ c2 vector v2 +…+ cp vector vp = vector 0 has only the trivial solution, c1 = … = cp = 0

Linear transformations

A transformation T is linear if:

i) T(vector u + vector v) = T(vector u) + T(vector v) for all vector u, vector v in the domain of T.

ii) T(c vector u) = cT(vector u) for all scalars c and all vector u in the domain of T.

one-to-one

A mapping T: R^n → R^m such that each vector b in R^m is the image of at most one vector x in R^n.

onto

A mapping T: R^n → R^m such that each vector b in R^m is the image of at least one vector x in R^n.

Matrix Product

If A is an mxn matrix and if B is an nxp matrix with columns vector b1,…,vector bp then the product AB is the mxp matrix whose columns are A vector b1,…,A vector bp.

Inverse of an nxn matrix A

An nxn matrix A^-1 such that AA^-1 = A^-1A = In