6 linea algebra

what is a linear equation

a straight line relationship between variables

what is a coefficient

a constant multiplying a variable

aka 4x (4 is the coefficient)

what is a constant term

e.g. (4x - 3y + 7 = 0) 7 is the constant

a constant with no variable

what is a variable (or unknown)

a quantity we are trying to find

what is a solution

a value that makes the equation true

what is a cartesian plane

the 2D coordinate system R²

What is 3D space

R³

what does a one variable linear equation look like

ax = b

what does a two variable linear equation look like

ax + by = c;

what does a three variable linear equation look like

ax + by + cz = d

what is a system of linear equations

several linear equations considered together; a solution must satisfy all of them simultaneously.

what is simultaneous equations

another name for a system

what is a coefficient matrix

the matrix of coefficients of the variables

what is an augmented matrix

the coefficient matrix with the constant terms append as an extra column

what is a consistent system

has at least one solution

what is an inconsistent system

has no solutions

what is a unique solution

exactly one solution

what is infinite solutions

infinitely many solutions

what is the standard form of two equations in two variables

ax1 + by1 = c1

ax2 + by2 = c2

What is substitution

Solve one equation for one variable, substitute into the other

What is elimination

Add or subtract scaled equations to cancel a variable

What is a matrix

A rectangular array of numbers arranged in rows and columns

What is a coefficient matrix

contains just the coefficients

What is an augmented matrix

Coefficient matrix + constants column

What is row echelon form

Staircase form with leading non-zero entries (often 1s) below which everything is 0

What is reduced row echelon form (RREF)

row echelon form with a leading 1 in each row, and zeros both above and below each leading 1

what is pivot (or leading one)

the first non-zero entry of a row

what is the guass-jordan algorithm

a systematic procedure for solving any linear system

how does the guass-jordan algorithm work

phase 1 — forward elimination

1. obtain a pivot in the leftmost column

2. subtract or add multiples of the pivot row from rows below to make all entries below the pivot zero

3. move to the next column to the right and repeat

phase 2 — back substitution

1. find the rightmost pivot — use it to eliminate all entries above it

2. move one column to the left and repeat