Math 1553 Introduction to Linear Algebra

Section 1.1

Solving systems of equations

Outline of Section 1.1

? Learn what it means to solve a system of linear equations ? Describe the solutions as points in Rn ? Learn what it means for a system of linear equations to be inconsistent

Solving equations

Solving equations

What does it mean to solve an equation? 2x = 10

x+y = 1

x+y+z =0

Find one solution to each. Can you find all of them?

A solution is a list of numbers (a.k.a. a vector). For example (3, -4, 1).

Solving equations

What does it mean to solve a system of equations?

x+y = 2 y=1

What about...

x+y+z =3 x+y-z =1 x-y+z =1

Is (1, 1, 1) a solution? Is (2, 0, 1) a solution? What are all the solutions? Soon, you will be able to see just by looking that there is exactly one solution.

Rn

Rn

R = denotes the set of all real numbers

Geometrically, this is the number line.

-3 -2 -1

0

1

2

3

n

R

=

all

ordered

n-tuples

(or

lists)

of

real

numbers

(x1,

x2,

x3,

...,

xn)

Solutions to systems of equations are exactly points in Rn. In other words, Rn is where our solutions will lie (the n depends on the system of equations).

We say Rn instead of R2 or R3 because many of the things we learn this semester work just as well for R777 as they do for R2 and R3. So when we say Rn we are talking about all of these at once. That is power!

Rn

When n = 2, we can visualize of R2 as the plane. (1, 2)

(0, -3)

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