: all real numbers

: all where x,y are real

: all where x,y,z are real

often visualized in a geometric space

: all where

Q: Consider , the xy-plane in , i.e. the set of all where x,y are real

is equal to ?

no, its a plane within

Every point in has 3 coordinates where

lives in , it is like but isn't equal to

# Lines, Planes in

This is the implicit form of a line

We could parametrize it as

**Linear Equations**

every variable appears separately and to the first power

**NOT Linear**

Solutions to linear systems ↔ intersection of lines, planes, etc in

(linear algebra) ↔ (geometry)

## Ex

Solutions to:

2 equations

3 variables

↔

Points in intersectoin of two planes

both in

Intersect in a line

Possibilities for the solution set of two linear equations in two unknowns generalizes to __arbitrary__ linear systems (ex: 17 linear equations in 13 variables)

- no solutions
- system is
**inconsistent** - exactly one solution
- system is
**consistent** - infinitely many solutions
- system is
**consistent**

# Augmented Matrices

Make life easier

x y z

Subtract from

# Solving Systems

**Valid operations:**

- scale an equations by a non-zero constant
- add a multiple of one equation to another
- swap order

**For corresponding augmented matrix:**

**Row Operations**

- Scale: Scale row by non-zero constant
- Row replacement: add multiple of one row to another
- Swap: swap two rows

**Goal:**Use row operations to "simplify" matrix into "solved" form

## Example 1

Solve

Try to remove the x term so we can solve with just y and z:

Swap and

Can back-solve from here using algebra

will continue with matrices

destroy what is above each box term or "pivot" term

removing z from other rows

trying to find the simplest way to describe a matrix

## Example 2

A linear system isinconsistent↔ the rightmost column of its augmented matrix is a pivot column