Linear Equations

what is the method for solving a system of linear equations?

go from top to bottom, variable by variable, so that each equation has only one variable

when does a linear system have infinitely many solutions?

when there are two leading variables and one free variable; when the intersection is a line

when does a linear system have no solutions?

when we have a false equation (e.g. 0 = c) in the system

how do we notate the solution of a linear system with infinitely many solutions

parametrize the free variable

matrix

rectangular array of numbers representing a system of equations

when are two matricies equal

they have the same size and corresponding entries

square matrix

n = m

diagonal matrix

a_ij = 0 when i =/= j

upper triangular matrix

a_ij = 0 when i > j

lower triangular matrix

a_ij = 0 when i < j

identity matrix (I_n)

all entries are 0s except for the 1s in the diagnoal

zero matrix

all 0 entries

(column) vector

matrix with one column

components

entries of a vector

row vector

matrix with one row

coefficient matrix

representation of a linear system by only the variable coefficients

augmented matrix

representation of a linear system by coefficients and constant which they equal to

what are the three properties that make linear systems easy to solve

P1: leading coefficient is 1 in each equation; P2: leading variable does not appear i other equations; P3: leading variables are in the “natural order”

what are the steps for solving a system of linear equations from top to bottom

1. divide equation so leading coefficient is 1

2. eliminate leading variable form all other equations

3. proceed likewise for other equations

4. Rearrange equations in natural order

5. solve each equation for leading variable

what are the three characteristics of reduced row-echelon form (rref)?

1. if a row has nonzero entries, the first nonzero coefficient is the leading 1 (pivot)

2. if a column contains a leading 1, all other entries are 0

3. If a row contains a leading 1, then each row alone contains a leading 1 further to the left

elementary row operations

bring a matrix to rref

what are the elementary row operations

1. divide by a nonzero scalar

2. subtract a multiple of a row from another row

3. swap two rows

a system of equations is consistent with

at least one solution

a system of equation is inconsistent

if and only if the rref form of its augmented matrix contains 0=1

rank of a matrix

number of leading 1’s in its rref

if a linear system has exactly one solution, then

the number of variables (m) must be less than or equal to the number of linear equations (n)

a linear system where the n is less than m has

either no solutions or infinitely many solutions

a linear system has a unique solution if and only if the rank of its coefficient matrix is

n

vector b is a linear combination of v if there exists

a scalar x such that b=xv

matrix form of a linear system

Ax=b