LA: 2.1-2.3 Terms

diagonal entries

entries with equal row and column index #s

main diagonal

The entries with equal row and column index #s.

diagonal matrix

A square matrix whose nondiagonal entries are all zero

zero matrix

a matrix in which every element is zero

matrices are equal

if the sizes are equal and corresponding entries are equal

sum A + B

the m x n matrix whose columns are the sums of the corresponding columns in A and B

scalar multiple of a vector

r is a scalar and A is a matrix whose columns are r times the corresponding columns in A

If A is an m x n matrix and if B is an n x p matrix with columns b_1, ... , b_p, then the product AB is

the m x p matrix whose columns are Ab_1 , ... , Ab_p

AB = A[b_1 b_2 ... b_p] = [Ab_1 Ab_2 ... Ab_p]

A and B commute

if AB = BA

transpose of A

Given an m x n matrix, it's the n x m matrix, denoted by A^T, whose columns are formed from the corresponding rows of A

invertible

An n x n matrix A if there's an n x n matrix C s. t. CA = I and AC = I where I = I_n, the n x n identity matrix

inverse matrix

the matrix when multiplied by the original matrix results in the identity matrix

AA^(-1) = (A^(-1))(A) = I_n

singular matrix

a matrix that is not invertible

nonsingular matrix

a matrix that's invertible

determinant of A

detA = ad - bc

elementary matrix

a matrix obtained by performing a single elementary row operation on an identity matrix

invertible linear transformation

A linear transformation T : R^n -> R^n is said to be invertible if there exists a function S : R^n -> R^n such that

S(T(x)) = x for all x in R^n

T(S(x)) = x for all x in R^n