Introduction to Linear Algebra

"Algebraic Method" of solving linear equations using Matrices.

Consider the following system:

3x1 + 2x2 - x3 + x4 =  -1

2x1          - x3 + 2x4 = 0

3x1 + x2 - 2x3 + 5x4 = -1

Augmented Matrix :

[ 3 2 -1 1 | -1 ]

[ 2 0 -1 2 |   0 ]

[ 3 1   2 5 |   2 ]

Coefficient Matrix :

[ 3 2 -1 1 ]

[ 2 0 -1 2 ]

[ 3 1   2 5 ]

Constant Matrix :

[  -1 ]

[   0 ]

[   2 ]

Properties of Real Numbers

Commutative Properties

• a + b = b + a
• Example: 7 + 3 = 3 + 7
• When we add two numbers, order doesn't matter.

• ab = ba
• Example: 3 * 5 = 5 * 3
• When we multiply two numbers, order doesn't matter.

Associative Properties

• (a + b) + c = a + (b + c)
• Example: (2 + 4) + 8 = 2 + (4 + 8)
• When we add three numbers, it does not matter which two we add first.

• (ab)c = a(bc)
• Example: (3 * 7) * 5 = 3 * (7 * 5)
• When we multiple three numbers, it does not matter which two we multiple first.

Distributive Property

• a(b + c) = ab + ac
• Example: 2 * (3 + 5) = 2 * 3 + 2 * 5
• When we multiply a number by a sum of two numbers, we get the same result as multiplying the number by each of the terms and then adding the products.
• (b + c)a = ab + ac
• Example: (4 + 5) * 2 = 2 * 4 + 2 * 5

Patterns of PI

When plotting graphs using radians I often found it tedious to have to try and memorize the formulas for degrees based on Pi.  However, it is not necessary to do so, if you can understand the fundamental concepts and are receptive to the continuing patterns throughout, charting graphs based on PI can be easy.

Watch the patterns !

π/2 , π, 3π/2, 2π, 5π/2, 3π, 7π/2, 4π, 9π/2, 5π, 11π/2, 6π, 13π/2, 7π, 15π/2, 8π

90, 180, 270, 360, 450, 540, 630, 720, 810, 900, 990, 1080, 1170, 1260, 1350, 1440 respectively. (degrees)

 Now let's incorporate that pattern into a similar pattern, this one including the special angles 30 and 60.

π/6, π/3, π/2, 2π/3, 5π/6, π, 7π/6, 4π/3, 3π/2, 5π/3, 11π/6, 2π, 13π/6, 7π/3, 5π/2, 8π/3, 17π/6, 3π

30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540 respectively. (degrees)