Sometimes we will use the notation 4=[ay]to indicate that 4 is the matrix whose iith element is ay. DEFINITION Two matrices A =[al and B=[bl are equal if and only if 4 and B are of the same order and for all i and j,ay=by. For example,if and then A B if and only ifx 1,y 2,w=3,and z =4. Vectors Any matrix with only one column (that is,any mX I matrix)may be thought of as a column vector.The number of rows in a column vector is the dimension of the column vector.Thus, 「1 2 may be thought of as a 2 X 1 matrix or a two-dimensional column vector.R"will denote the set of all m-dimensional column vectors. In analogous fashion,we can think of any vector with only one row(a 1 X n matrix as a row vector.The dimension of a row vector is the number of columns in the vector.Thus, [9 2 3]may be viewed as a 1 X 3 matrix or a three-dimensional row vector.In this book, vectors appear in boldface type:for instance,vector v.An m-dimensional vector(either row or column)in which all elements equal zero is called a zero vector (written 0).Thus, 0 [00] and 0 are two-dimensional zero vectors. Any m-dimensional vector corresponds to a directed line segment in the m-dimensional plane.For example,in the two-dimensional plane,the vector 17 2 corresponds to the line segment joining the point 01 to the point 周 The directed line segments corresponding to u-卧v=[w=[2 are drawn in Figure 1.Sometimes we will use the notation A [aij] to indicate that A is the matrix whose ijth element is aij. DEFINITION ■ Two matrices A [aij] and B [bij] are equal if and only if A and B are of the same order and for all i and j, aij bij. ■ For example, if A and B then A B if and only if x 1, y 2, w 3, and z 4. Vectors Any matrix with only one column (that is, any m 1 matrix) may be thought of as a column vector. The number of rows in a column vector is the dimension of the column vector. Thus, may be thought of as a 2 1 matrix or a two-dimensional column vector. Rm will denote the set of all m-dimensional column vectors. In analogous fashion, we can think of any vector with only one row (a 1 n matrix as a row vector. The dimension of a row vector is the number of columns in the vector. Thus, [9 2 3] may be viewed as a 1 3 matrix or a three-dimensional row vector. In this book, vectors appear in boldface type: for instance, vector v. An m-dimensional vector (either row or column) in which all elements equal zero is called a zero vector (written 0). Thus, [0 0] and are two-dimensional zero vectors. Any m-dimensional vector corresponds to a directed line segment in the m-dimensional plane. For example, in the two-dimensional plane, the vector u corresponds to the line segment joining the point to the point The directed line segments corresponding to u , v , w are drawn in Figure 1. 1 2 1 3 1 2 1 2 0 0 1 2 0 0 1 2 y z x w 2 4 1 3 12 CHAPTER 2 Basic Linear Algebra