mat 2310 0607_2_note0_2

mat 2310 0607_2_note0_2 - Lecture Note 0 Dr Jeff Chak-Fu...

This preview shows pages 1–7. Sign up to view the full content.

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

View Full Document

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture Note 0: Jan 10, 2007 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310c Linear Algebra and Its Applications Spring, 2007 Produced by Jeff Chak-Fu WONG 1 L INEAR E QUATIONS AND M ATRICES 1. Linear Systems 2. Matrices 3. Dot Product and Matrix Multiplication 4. Properties of Matrix Operations 5. Solutions of Linear Systems of Equations 6. The Inverse of A Matrix 7. LU-Factorization LINEAR EQUATIONS AND MATRICES 2 P ROPERTIES OF M ATRIX O PERATIONS 1. Properties of Matrix Addition 2. Properties of Matrix Multiplication • Definition of Identity Matrix • Power of A Matrix 3. Properties of Scalar Multiplication 4. Properties of Transpose • Definition of Symmetric Matrix PROPERTIES OF MATRIX OPERATIONS 3 P ROPERTIES OF M ATRIX A DDITION Theorem 0.1 (Properties of Matrix Addition) Let A,B,C and D be m × n matrices. (a) A + B = B + A (Commutative law for addition) (b) A + ( B + C ) = ( A + B ) + C. (Associate law for addition) (c) There is a unique m × n matrix O such that A + O = A (1) for any m × n matrix A . The matrix O is called the m × n additive identity or zero matrix. PROPERTIES OF MATRIX ADDITION 4 (d) For each m × n matrix A , there is a unique m × n matrix D such that A + D = O. (2) We shall write D as (- A ), so that (2) can be written as A + (- A ) = O. The matrix (- A ) is called the additive inverse or negative of A . Proof (a) To establish (a), we must prove that the i,j th element of A + B equals the i,j th element of B + A . The i,j th element of A + B is a ij + b ij ; the i,j th element of B + A . The i,j th element of B + A is b ij + a ij . Since the elements PROPERTIES OF MATRIX ADDITION 5 a ij are real (or complex) numbers, a ij + b ij = b ij + a ij (1 ≤ i ≤ m, 1 ≤ j ≤ n ) , the result follows. (b) Exercise. (c) Let U = [ u ij ] . Then A + U = A if and only if a ij + u ij = a ij , which holds if and only if u ij = 0 . Thus U is the m × n matrix all of whose entries are zero; U is denoted by O . (d) Exercise....
View Full Document

{[ snackBarMessage ]}

Page1 / 29

mat 2310 0607_2_note0_2 - Lecture Note 0 Dr Jeff Chak-Fu...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online