"This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications." – Electric Review
A comprehensive introduction, Linear Algebra: Ideas and Applications, Fourth Edition provides a discussion of the theory and applications of linear algebra that blends abstract and computational concepts. With a focus on the development of mathematical intuition, the book emphasizes the need to understand both the applications of a particular technique and the mathematical ideas underlying the technique. The book introduces each new concept in the context of an explicit numerical example, which allows the abstract concepts to grow organically out of the necessity to solve specific problems. The intuitive discussions are consistently followed by rigorous statements of results and proofs. Linear Algebra: Ideas and Applications, Fourth Edition also features: Two new and independent sections on the rapidly developing subject of wavelets A thoroughly updated section on electrical circuit theory Illuminating applications of linear algebra with self-study questions for additional study End-of-chapter summaries and sections with true-false questions to aid readers with further comprehension of the presented material Numerous computer exercises throughout using MATLAB® code
Linear Algebra: Ideas and Applications, Fourth Edition is an excellent undergraduate-level textbook for one or two semester courses for students majoring in mathematics, science, computer science, and engineering. With an emphasis on intuition development, the book is also an ideal self-study reference.
SYSTEMS OF LINEAR EQUATIONS
1.1 The Vector Space of m × n Matrices
It is difficult to go through life without seeing matrices. For example, the 2014 annual report of Acme Squidget might contain the Table 1.1 , which shows how much profit (in millions of dollars) each branch made from the sale of each of the company's three varieties of squidgets in 2014.
TABLE 1.1 Profits: 2014
Red Blue Green Total Kokomo 11.4 5.7 6.3 23.4 Philly 9.1 6.7 5.5 21.3 Oakland 14.3 6.2 5.0 25.5 Atlanta 10.0 7.1 5.7 22.8 Total 44.8 25.7 22.5 93.0
If we were to enter this data into a computer, we might enter it as a rectangular array without labels. Such an array is called a matrix . The Acme profits for 2014 would be described by the following matrix. This matrix is a 5 × 4 matrix (read "five by four") in that it has five rows and four columns. We would also say that its "size" is 5 × 4. In general, a matrix has size m × n if it has m rows and n columns.
Definition 1.1 The set of all m × n matrices is denoted M ( m , n ).
Each row of an m × n matrix may be thought of as a 1 × n matrix. The rows are numbered from top to bottom. Thus, the second row of the Acme profit matrix is the 1 × 4 matrix
This matrix would be called the "profit vector" for the Philly branch. (In general, any matrix with only one row is called a row vector . For the sake of legibility, we usually separate the entries in row vectors by commas, as above.)
Similarly, a matrix with only one column is called a column vector . The columns are numbered from left to right. Thus, the third column of the Acme profit matrix is the column vector
This matrix is the "green squidget profit vector."
If A 1, A 2, ..., An is a sequence of m × 1 column vectors, then the m × n matrix A that has the Ai as columns is denoted
Similarly, if B 1, B 2, ..., Bm is a sequence of 1 × n row vectors, then the m × n matrix B that has the Bi as rows is denoted
In general, if a matrix is denoted by an uppercase letter, such as A , then the entry in the i th row and j th column may be denoted by either Aij or aij , using the corresponding lowercase letter. We shall refer to aij as the "( i , j ) entry of A ." For example, for the matrix P above, the (2, 3) entry is p 23 = 5.5. Note that the row number comes first. Thus, the most general 2 × 3 matrix is
We will also occasionally write " A = [ aij ]," meaning that " A is the matrix whose ( i , j ) entry is aij ."
At times, we want to take data from two tables, manipulate it in some manner, and display it in a third table. For example, suppose that we want to study the performance of each division of Acme Squidget over the two-year period, 2013-2014. We go back to the 2013 annual report, finding the 2013 profit matrix to be
If we want the totals for the two-year period, we simply add the entries of this matrix to the corresponding entries from the 2014 profit matrix. Thus, for example, over the two-year period, the Kokomo division made 5