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If you were asked for your weight in pounds, you would use a real number
such as 140 to answer the question. If you were asked for your height in
inches, you would answer with another real number such as 66.5. If we asked
these questions to everyone in the class, we would want some way to know
which weight goes with which height. One way to organize this data is to use
an ordered pair. We could represent your weight and height with the ordered
pair (140, 66.5). This is called an ordered pair
because we always list the information in the same order. In other words, we
list weight first and then height in every pair of numbers, so (140, 66.5)
would be different from (66.5, 140). The elements
are the individual pieces of information. Elements are also referred to as
entries or components. In this book, we will only use real numbers as
elements. The elements of this ordered pair are 140 and 66.5. We could also
ask you for your age in years and append that information so that we have
the ordered triple (140, 66.5, 18). We could ask you for n pieces of
information, where n is any counting number. If we arrange the n pieces
of information in a specific order, we call it an ordered n-tuple. In
general, lists of ordered information are called vectors. If we write them
in rows, as we did above, we call them row vectors. If we write them in
columns, such as
 and
we call them column vectors. Definition 1.1 A real n-vector is an ordered n-tuple of real numbers.Since we are only working with real numbers in this book, we will drop the word real when referring to vectors. When it is not important to specify how many elements are in the vector, we drop the qualifier n. Remark 1 Did you notice that we used parentheses on some vectors and brackets on others? Actually, both are accepted notations, but we will use brackets for consistency throughout the rest of the book. Remark 2 Sometimes you will see the elements of a row vector separated by commas. Commas are not necessary unless confusion can arise without the use of commas.
If you were asked to add, subtract, or multiply real numbers, you would know
what to do. If we are going to use vectors to help us organize our
information, we also need rules for vectors so that when we add, subtract,
or multiply vectors, we get the same solutions as if we had not organized
our data this way. Remember that vectors are simply tools that we use to
display information in an organized manner. Therefore, we do not want our
solutions to change just because we organized our data into a vector. As we
study this book, we will learn more about how to perform mathematical
operations with vectors. The Cardinals win seven, lose six, and tie one. The Eagles win five, lose eight, and tie one. The Falcons win two, lose twelve, and have no ties. The Owls win nine, lose five, and have no ties.We can represent this data using the four vectors [7 6 1 ], [5 8 1 ], [2 12 0 ], and [9 5 0 ]. However, it would be nice if we could combine all these vectors together into one set of data. If we consider each vector as one row of an array, then we will have all our data in one arrangement. Definition 1.2 A real matrix is an arrangement of real numbers into rows and columns.Since we are only working with real numbers in this book, we also will drop the word real when referring to matrices. Notice that a vector is a special matrix that has only one row or one column. When we organize our vectors into a matrix, it could look like this:
 Basically, we have all of our information organized into one arrangement called a matrix. We have all the appropriate numbers in our matrix, but if we want to know which numbers correspond to which team, we have to look back at our paragraph. For this reason, we often label our matrices. Labels are not a formal part of the matrix, but they are very useful. Our matrix could look like this after it has been labeled:  Just by looking at this matrix, we can tell that the Owls won the most games and the Falcons lost the most. One of the advantages of matrices is that information is easier to see and compare than when it is not organized into a matrix. This matrix is referred to as a 4 by 3 matrix (often written 4x3) because there are 4 rows and 3 columns. Therefore, the dimensions of this matrix are 4 by 3. The dimensions of a matrix tell you the "size" of the matrix because they tell you the number of rows and columns in the matrix. By convention, we list the number of rows before the number of columns. Definition 1.3 The dimensions of a matrix are the number of rows and columns (listed in that order) of the matrix.Each element of the matrix is named according to its position. Typically, capital letters represent matrices and small letters with subscripts represent elements in the matrix. Since vectors can be considered to be matrices with only one row or one column, they could be labeled with capital letters also. However, vectors are usually represented by small letters. If we name the above matrix A, the element 6 is in the position a12 (read a one two) because it is in row 1 and column 2. Also by convention, we list the row number of the element before the column number. An element in row i and column j would be denoted by aij. This gives us a compact way to refer to specific elements of a matrix. Remark 3 Although some mathematicians make a distinction between a 1 by 1 matrix, a 1-vector, and a real number, we will not make any distinction between them and will treat them exactly the same. Can you represent the same information as before in a 3 by 4 matrix? Yes, you can. It would look like the matrix B which follows.
Matrix B is the transpose of A, and A is the transpose of B. Transposing a matrix results in writing the columns as rows and the rows as columns, but what really happens is that element aij is placed in the position bji of the new matrix. Therefore, a12 moves to the position b21 when we form the transpose of A. The transpose of A is denoted by AT (read A transpose). Therefore, matrix B is AT.
Definition 1.4 By the transpose of the m by n matrix A, denoted by AT, we mean the n by m matrix which has aji as its (i,j)th element. Definition 1.5 We say that two m by n matrices, A and B are equal if their corresponding elements are equal.In other words, A = B if A and B have the same dimensions and a11 = b11, a12 = b12, etc. Is A = AT? Usually not, but we have a special word for a matrix which satisfies A = AT. Definition 1.6 A matrix is said to be symmetric if A = AT.Observe that the following matrix is symmetric:
Notice that aij = aji for all i and j; as is true for all symmetric matrices. Symmetric matrices are easy to spot because if you draw a line down the main diagonal (from 9 to 3 in this matrix), then the two halves are mirror images of each other. Symmetric matrices have many special qualities that will be used when you study matrices in more detail. The matrix S, given above, has another special property; it is a square matrix because S has the same number of rows as columns. Notice that S is a 4 by 4 square matrix. We said that the main diagonal for S runs from 9 to 3. For any square matrix, the main diagonal runs from the upper left corner to the lower right corner. Definition 1.7 We say that an m by n matrix is square if m = n. |
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