{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

la02-m

# la02-m - LINEAR ALGEBRA W W L CHEN c W W L Chen 1982 2005...

This preview shows pages 1–3. Sign up to view the full content.

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2005. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 2 MATRICES 2.1. Introduction A rectangular array of numbers of the form (1) a 11 . . . a 1 n . . . . . . a m 1 . . . a mn is called an m × n matrix, with m rows and n columns. We count rows from the top and columns from the left. Hence ( a i 1 . . . a in ) and a 1 j . . . a mj represent respectively the i -th row and the j -th column of the matrix (1), and a ij represents the entry in the matrix (1) on the i -th row and j -th column. Example 2.1.1. Consider the 3 × 4 matrix 2 4 3 1 3 1 5 2 1 0 7 6 . Here (3 1 5 2) and 3 5 7 Chapter 2 : Matrices page 1 of 19

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Linear Algebra c W W L Chen, 1982, 2005 represent respectively the 2-nd row and the 3-rd column of the matrix, and 5 represents the entry in the matrix on the 2-nd row and 3-rd column. We now consider the question of arithmetic involving matrices. First of all, let us study the problem of addition. A reasonable theory can be derived from the following definition. Definition. Suppose that the two matrices A = a 11 . . . a 1 n . . . . . . a m 1 . . . a mn and B = b 11 . . . b 1 n . . . . . . b m 1 . . . b mn both have m rows and n columns. Then we write A + B = a 11 + b 11 . . . a 1 n + b 1 n . . . . . . a m 1 + b m 1 . . . a mn + b mn and call this the sum of the two matrices A and B . Example 2.1.2. Suppose that A = 2 4 3 1 3 1 5 2 1 0 7 6 and B = 1 2 2 7 0 2 4 1 2 1 3 3 . Then A + B = 2 + 1 4 + 2 3 2 1 + 7 3 + 0 1 + 2 5 + 4 2 1 1 2 0 + 1 7 + 3 6 + 3 = 3 6 1 6 3 3 9 1 3 1 10 9 . Example 2.1.3. We do not have a definition for “adding” the matrices 2 4 3 1 1 0 7 6 and 2 4 3 3 1 5 1 0 7 . PROPOSITION 2A. (MATRIX ADDITION) Suppose that A, B, C are m × n matrices. Suppose further that O represents the m × n matrix with all entries zero. Then (a) A + B = B + A ; (b) A + ( B + C ) = ( A + B ) + C ; (c) A + O = A ; and (d) there is an m × n matrix A such that A + A = O . Proof. Parts (a)–(c) are easy consequences of ordinary addition, as matrix addition is simply entry-wise addition. For part (d), we can consider the matrix A obtained from A by multiplying each entry of A by 1. The theory of multiplication is rather more complicated, and includes multiplication of a matrix by a scalar as well as multiplication of two matrices.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 19

la02-m - LINEAR ALGEBRA W W L CHEN c W W L Chen 1982 2005...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online