Addition of Matrices
Denote the sum of two matrices A and B (of the same dimensions) by C = A + B.. The sum is defined by adding entries with the same indices
[pic]
over all i and j.
Example:
[pic]
Subtraction of Matrices
Subtraction is performed in analogous way.
Example:
[pic]
Scalar multiplication
To multiply a matrix with a real number, we multiply each element with this number.
Example:
[pic]
Multiplication of a row vector by a column vector
This multiplication is only possible if the row vector and the column vector have the same number of elements. To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example:
[pic]
If the row vector and the column vector are not of the same length, their product is not defined.
Example:
[pic]
The Product of a Row Vector and Matrix
When the number of elements in row vector is the same as the number of rows in the second matrix then this matrix multiplication can be performed.
Example:
[pic]
If the number of elements in row vector is NOT the same as the number of rows in the second matrix then their product is not defined.
Example:
[pic]

Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization).

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists.

Standard form
Standard form is the usual and most intuitive form of describing a linear programming problem. It consists of the following three parts: • A linear function to be maximized
e.g. [pic] • Problem constraints of the following form
e.g.
[pic] • Non-negative variables
e.g.
[pic]
The problem is usually expressed in matrix form, and then becomes:
[pic]
Other forms, such as minimization problems, problems with constraints on alternative forms, as well as problems involving negative variables can always be rewritten into an equivalent problem in standard form.
Example
Suppose that a farmer has a piece of farm land, say L km2, to be planted with either wheat or barley or some combination of the two. The farmer has a limited amount of fertilizer,F kilograms, and insecticide, P kilograms. Every square kilometer of wheat requires F1 kilograms of fertilizer and P1 kilograms of insecticide, while every square kilometer of barley requires F2 kilograms of fertilizer and P2 kilograms of insecticide. Let S1 be the selling price of wheat per square kilometer, and S2 be the selling price of barley. If we denote the area of land planted with wheat and barley by x1 and x2 respectively, then profit can be maximized by choosing optimal values for x1 and x2. This problem can be expressed with the following linear programming problem in the standard form:
|Maximize: [pic] |(maximize the revenue—revenue is the "objective function") |
|Subject to: |[pic] |(limit on total area) |
| |[pic] |(limit on fertilizer) |
| |[pic] |(limit on insecticide) |
| |[pic] |(cannot plant a negative area). |

Which in matrix form becomes: maximize [pic] subject to [pic]
Linear Equations

A linear equation is an equation for a straight line

These are all linear equations:

|y = 2x+1 |5x = 6+3y |y/2 = 3 - x |

Let us look more closely at one example:

Example: y = 2x+1 is a linear equation:

[pic]

The graph of y = 2x+1 is a straight line

• When x increases, y increases twice as fast, hence 2x

• When x is 0, y is already 1. Hence +1 is also needed

• So: y = 2x + 1

Here are some example values:

|x |y = 2x + 1 |
|-1 |y = 2 × (-1) + 1 = -1 |
|0 |y = 2 × 0 + 1 = 1 |
|1 |y = 2 × 1 + 1 = 3 |
|2 |y = 2 × 2 + 1 = 5 |

Check for yourself that those points are part of the line above!

Different Forms

There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").

Examples: These are linear equations:

|[pic| |y = 3x - 6 |
|] | | |
|[pic| |y - 2 = 3(x + |
|] | |1) |
|[pic| |y + 2x - 2 = 0 |
|] | | |
|[pic| |5x = 6 |
|] | | |
|[pic| |y/2 = 3 |
|] | | |

But the variables (like "x" or "y") in Linear Equations do NOT have:

• Exponents (like the 2 in x2)

• Square roots, cube roots, etc

Examples: These are NOT linear equations:

|[pic| |y2 - 2 = 0 |
|] | | |
|[pic| |3√x - y = 6|
|] | | |
|[pic| |x3/2 = 16 |
|] | | |

Slope-Intercept Form

The most common form is the slope-intercept equation of a straight line:

|[pic] | |[pic] |
| | | |
| | |Slope (or Gradient) |
| | |Y Intercept |
| | | |
| | | |

Example: y = 2x + 1

(Our example from the top, which is in Slope-Intercept form)

• Slope: m = 2

• Intercept: b = 1

|[pic] | |Play With It ! |
| | |You can see the effect of different values of m and b atExplore the Straight Line Graph |

Point-Slope Form

Another common one is the Point-Slope Form of the equation of a straight line:

|y - y1 = m(x - | |[pic] |
|x1) | | |

Example: y - 3 = ¼(x - 2)

• x1 = 2

• y1 = 3

• m = ¼

General Form

And there is also the General Form of the equation of a straight line:

|Ax + By + C = 0 |
|(A and B cannot both be 0) |

Example: 3x + 2y - 4 = 0

• A = 3

• B = 2

• C = -4

There are other, less common forms as well.

As a Function

Sometimes a linear equation is written as a function, with f(x) instead of y:

|y = 2x - 3 |f(x) = 2x - 3 |
|These are the same! |

And functions are not always written using f(x):

|y = 2x - 3 |w(u) = 2u - 3 |h(z) = 2z - 3 |
|These are also the same! |

The Identity Function

There is a special linear function called the "Identity Function":

f(x) = x

And here is its graph:

[pic]
It makes a 45° (its slope is 1)

It is called "Identity" because what comes out is identical to what goes in:

|In |Out |
|0 |0 |
|5 |5 |
|-2 |-2 |
|...etc |...etc |

Constant Functions

Another special type of linear function is the Constant Function ... it is a horizontal line:

[pic]

f(x) = C

No matter what value of "x", f(x) is always equal to some constant value.

Using Linear Equations

You may like to read some of the things you can do with lines:

• Finding the Midpoint of a Line Segment

• Finding Parallel and Perpendicular Lines

• Finding the Equation of a Line from 2 Points

PROPERTIES OF TRANSPOSE MATRICES

• f r is a scalar element and A and B represent matrices:

•

• [pic]

• shows that the transpose of a transpose matrix is the original matrix

•

• Example

•

• [pic]

• shows that the transpose of two added matrices is the same as the addition of the two transpose matrices

•

• Example

•

• [pic]

• shows that when a scalar element is multiplied to a matrix, the order of transposition is irrelevant

•

• Example

•

• [pic]

• shows that the transpose of a product of matrices equals the product of their transposes in reverse order