Matrix

In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. An example of a matrix with 2 rows and 3 columns is



Matrices of the same size can be added or subtracted element by element. The rule for matrix multiplication is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second.
A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation. If R is a rotation matrix and v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two matrices is a matrix that represents the composition of two linear transformations.
Another application of matrices is in the solution of a system of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Eigenvalues and eigenvectors provide insight into the geometry of linear transformations.

Matrices find applications in most scientific fields. In physics, matrices are used to study electrical circuits, optics, and quantum mechanics. In computer graphics, matrices are used to project a 3-dimensional image onto a 2-dimensional screen, and to create realistic-seeming motion. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.

A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to the structure of particular matrix structures, e.g. sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example is the matrix representing the derivative operator, which acts on the Taylor series of a function.

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Determinant

The determinant det(A) or |A| of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero.
Its absolute value equals the area (in R2) or volume (in R3) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.

The determinant of 2-by-2 matrices is given by

   

When the determinant is equal to one, then the matrix represents an equi-areal mapping. The determinant of 3-by-3 matrices involves 6 terms (rule of Sarrus). The more lengthy Leibniz formula generalises these two formulae to all dimensions.

The determinant of a product of square matrices equals the product of their determinants
: det(AB) = det(A) · det(B).
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1.
Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix.
 Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices. This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1-by-1 matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula.
 Determinants can be used to solve linear systems using Cramer's rule, where the division of the determinants of two related square matrices equates to the value of each of the system's variables.

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Graham's number

Graham's number

, named after Ronald Graham, is a large number often described as the largest finite number that has ever been seriously used in a mathematical proof. Guinness World Records even listed Graham's number as the World Champion largest number. Graham's number is much larger than other well known large numbers such as a googol and a googolplex, and even larger than Skewes' number and Moser's number, other well-known large numbers. Indeed, there is no concise way to write Graham's number, or any reasonable approximation, using conventional mathematical operators. Even power towers (of the form a ^{ b ^{ c ^{ cdot ^{ cdot ^{ cdot}}}}}) are useless for this purpose. It can be most easily notated by recursive means using Knuth's up-arrow notation or the Hyper operator.



Graham's number is connected to the following problem in the branch of mathematics known as Ramsey theory:

    Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2n vertices. Then colour each of the edges of this graph using only the colours red and black. What is the smallest value of n for which every possible such colouring must necessarily contain a single-coloured complete sub-graph with 4 vertices which lie in a plane?

Although the solution to this problem is not yet known, Graham's number is the smallest known upper bound for it. This bound was found by Graham and B. L. Rothschild (see (GR), corollary 12). They also provided the lower bound 6, adding the qualifying understatement: "Clearly, there is some room for improvement here."


In Penrose Tiles to Trapdoor Ciphers, Martin Gardner wrote, "Ramsey-theory experts believe the actual Ramsey number for this problem is probably 6, making Graham's number perhaps the worst smallest-upper-bound ever discovered." More recently Geoff Exoo of Indiana State University has shown (in 2003) that it must be at least 11 and provided evidence that it is larger.

Definition of Graham's number

Using Knuth's up-arrow notation, Graham's number G is defined as

Equivalently,

G = g₆₄ where ,

or

G = f⁶⁴(4) where f(n) = hyper(3,n + 2,3) and hyper() is the hyper operator.

Graham's number G itself cannot succinctly be expressed in Conway chained arrow notation, but , see bounds on Graham's number in terms of Conway chained arrow notation.

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Two-Step Equations

Two-Step Equations

Solve a Two-Step Equation

We’ve seen how to solve for an unknown by isolating it on one side of an equation and then evaluating the other side. Now we’ll see how to solve equations where the variable takes more than one step to isolate.

Example 1

Rebecca has three bags containing the same number of marbles, plus two marbles left over. She places them on one side of a balance. Chris, who has more marbles than Rebecca, adds marbles to the other side of the balance. He finds that with 29 marbles, the scales balance. How many marbles are in each bag? Assume the bags weigh nothing.

Solution
We know that the system balances, so the weights on each side must be equal. If we use to represent the number of marbles in each bag, then we can see that on the left side of the scale we have three bags (each containing marbles) plus two extra marbles, and on the right side of the scale we have 29 marbles. The balancing of the scales is similar to the balancing of the following equation.

“Three bags plus two marbles equals 29 marbles”
To solve for , we need to first get all the variables (terms containing an ) alone on one side of the equation. We’ve already got all the ’s on one side; now we just need to isolate them.

There are nine marbles in each bag.
We can do the same with the real objects as we did with the equation. Just as we subtracted 2 from both sides of the equals sign, we could remove two marbles from each side of the scale. Because we removed the same number of marbles from each side, we know the scales will still balance.

Then, because there are three bags of marbles on the left-hand side of the scale, we can divide the marbles on the right-hand side into three equal piles. You can see that there are nine marbles in each.
Three bags of marbles balances three piles of nine marbles.

So each bag of marbles balances nine marbles, meaning that each bag contains nine marbles.

Example 2

Solve .
This equation has the buried in parentheses. To dig it out, we can proceed in one of two ways: we can either distribute the six on the left, or divide both sides by six to remove it from the left. Since the right-hand side of the equation is a multiple of six, it makes sense to divide. That gives us . Then we can subtract 4 from both sides to get .

Example 3

Solve .
It’s always a good idea to get rid of fractions first. Multiplying both sides by 5 gives us , and then we can add 3 to both sides to get .

Example 4

Solve .
First, we’ll cancel the fraction on the left by multiplying by the reciprocal (the multiplicative inverse).

Then we subtract 1 from both sides. ( is equivalent to 1.)

These examples are called two-step equations, because we need to perform two separate operations on the equation to isolate the variable.

 

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Solving Equations Using Multiplication and Division

 Solving Equations Using Multiplication and Division

Suppose you are selling pizza for $1.50 a slice and you can get eight slices out of a single pizza. How much money do you get for a single pizza? It shouldn’t take you long to figure out that you get . You solved this problem by multiplying. Here’s how to do the same thing algebraically, using to stand for the cost in dollars of the whole pizza.

Example 4

Solve .
Our is being multiplied by one-eighth. To cancel that out and get by itself, we have to multiply by the reciprocal, 8. Don’t forget to multiply both sides of the equation.

Example 5

Solve .
is equivalent to , so to cancel out that , we multiply by the reciprocal, .

Example 6

Solve .
0.25 is the decimal equivalent of one fourth, so to cancel out the 0.25 factor we would multiply by 4.

Solving by division is another way to isolate . Suppose you buy five identical candy bars, and you are charged $3.25. How much did each candy bar cost? You might just divide $3.25 by 5, but let’s see how this problem looks in algebra.

Example 7

Solve .
To cancel the 5, we divide both sides by 5.

Example 8

Solve .
Divide both sides by 7.

Example 9

Solve .
Divide by 1.375

Notice the bar above the final two decimals; it means that those digits recur, or repeat. The full answer is 0.872727272727272...

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Solving Equations Using Addition and Subtraction

 

 

Solving Equations Using Addition and Subtraction

When we work with an algebraic equation, it’s important to remember that the two sides have to stay equal for the equation to stay true. We can change the equation around however we want, but whatever we do to one side of the equation, we have to do to the other side. In the introduction above, for example, we could get from the first equation to the second equation by subtracting 22 from both sides:

Similarly, we can add numbers to each side of an equation to help solve for our unknown.

Example 1

Solve .

Solution
To solve an equation for , we need to isolate that is, we need to get it by itself on one side of the equals sign. Right now our has a 3 subtracted from it. To reverse this, we’ll add 3—but we must add 3 to both sides.

Example 2

Solve

Solution
It doesn’t matter what the variable is—the solving process is the same.

Make sure you understand the addition of decimals in this example!

Example 3

Solve .
Solution
To isolate , we need to subtract from both sides.

Now we have to subtract fractions, which means we need to find the LCD. Since 5 and 7 are both prime, their lowest common multiple is just their product, 35.

Make sure you’re comfortable with decimals and fractions! To master algebra, you’ll need to work with them frequently.

 

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