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Introduction to Linear Algebra: Applications & Overview

Instructor: Glenda Boozer
What is linear algebra, anyway, and why should we care? What can we do with it? Let's look at some examples of the kinds of problems that can be made easier by the use of linear algebra techniques.

Definitions

Linear algebra is the branch of mathematics that deals with matrices, vectors and vector spaces, and systems of linear equations.

A vector is any quantity that is represented by both an amount and a direction, like velocity or force, like an arrow of a certain length pointing a certain way. A vector can be represented by an ordered pair, triple, or more, like (3, 2, -1). This would stand for an arrow pointing from (0, 0, 0) to (3, 2, -1). We can find the length and direction using the ordered triple (3, 2, -1).

A matrix (the plural is matrices) is like a table or array in which numbers are arranged in rows and columns. We can think of the rows of the matrix as vectors, and we can also think of the columns of a matrix as vectors.

A system of linear equations is probably familiar to you, at least to some degree. It consists of more than 1 equation in which none of the variables are raised to any power other than 1, and any solution of the system will have to make all of the equations true. Here's a simple example:

3x + 2y = 5

2x + 5y = 3

If we organize the equations so that the variables are in the same order, we can look at the coefficients (the numbers that are multiplied by the variables) and the constants (the numbers that are not multiplied by anything) as a matrix, like this:

3 2 5
2 5 3

It all fits together!

Applications

One of the things we can do with a vector is to find its length if we know its coordinates. For example, suppose we have a vector represented by (2,3):

Vector

We can find the length of this vector by using the Pythagorean Theorem: If c is the length of the hypoteneuse (the long side) of a right triangle and a and b are the lengths of the other sides, then c^2 = a^2 + b^2. In fact, the length of 'A' is the square root of 2^2 + 3^2, which is the square root of 13. We can extend this into 3 dimensions or more: to find the length of a vector in any number of dimensions, we square each coordinate, add up all those squares, and take the square root of that sum. Let's see it written out:

Length of a vector

The length of a vector can represent the speed of a moving object or the size of a force, or many other things other than simple distance.

Now, let's look at another system of equations, along with its augmented matrix, a matrix of all the numbers in the system. Suppose I need to create a mixture for dog food that has 8 grams of carbohydrate, 7 grams of protein, and 11 grams of fat. I have 3 raw ingredients: x has 1 gram of carbohydrate, 2 grams of protein, and 3 grams of fat per ounce; y has 2 grams of carbohydrate, 1 gram of protein, and 1 gram of fat per ounce; z has 1 gram of carbohydrate, 1 gram of protein, and 2 grams of fat per ounce. We can write this as a system of equations like this:

x + 2y + z = 8

2x + y + z = 7

3x + y + 2z = 11

1 2 1 8
2 1 1 7
3 1 2 11

We can solve the system of equations by performing elementary row operations on the matrix. There are 3 kinds of elementary row operations:

- Multiply a row through by a number that is not 0.

- Switch the positions of 2 rows.

- Add a multiple of 1 row to another.

Let's use this matrix to solve:

1 2 1 8
2 1 1 7
3 1 2 11

We already have a 1 in the x position, so let's leave it and try to get zeroes in that position for the others. We can subtract double the first row from the second row:

1 2 1 8
0 -3 -1 -9
3 1 2 11

Now we can subtract 3 times the first row from the third row:

1 2 1 8
0 -3 -1 -9
0 -5 -1 -13

We can get a 1 in the y position in the second row by dividing by -3 all the way across:

1 2 1 8
0 1 1/3 3
0 -5 -1 -13

Now we can subtract 2 times the second row from the first:

1 0 1/3 2
0 1 1/3 3
0 -5 -1 -13

We can also add 5 times the second row to the third row.

1 0 1/3 2
0 1 1/3 3
0 0 2/3 2

Now we can divide across the entire third row by 2/3.

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