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Lect1-04-web

# Lect1-04-web - MATH 304 Fall 2011 Linear Algebra Help...

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MATH 304, Fall 2011 Linear Algebra

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Help sessions: The Math 304/309/311/323 Help Session will be held on MW from 7:30-10:00pm and on TR from 6:30-9:00 p.m., BLOC 160 Homework assignment #2 (due Thursday, September 15) All problems are from Leon’s book (8th edition). Section 1.3: 1b, 1c, 1e, 1f, Section 1.4: 9 Section 1.5: 10c, 10e, 10h, 11a, 11b
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

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Matrices Defnition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 . . . a mn Notation: A = ( a ij ) 1 i n , 1 j m or simply A = ( a ij ) if the dimensions are known.
Matrix algebra: linear operations Addition: two matrices of the same dimensions can be added by adding their corresponding entries. Scalar multiplication: to multiply a matrix A by a scalar r , one multiplies each entry of A by r . Zero matrix O : all entries are zeros. Negative: A is deFned as ( 1) A . Subtraction: A B is deFned as A + ( B ). As far as the linear operations are concerned, the m × n matrices can be regarded as mn -dimensional vectors.

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Properties of linear operations ( A + B ) + C = A + ( B + C ) A + B = B + A A + O = O + A = A A + ( A ) = ( A ) + A = O r ( sA ) = ( rs ) A r ( A + B ) = rA + rB ( r + s ) A = rA + sA 1 A = A 0 A = O
Dot product Defnition. The dot product of n -dimensional vectors x = ( x 1 , x 2 , . . . , x n ) and y = ( y 1 , y 2 , . . . , y n ) is a scalar x · y = x 1 y 1 + x 2 y 2 + ··· + x n y n = n s k =1 x k y k . The dot product is also called the scalar product .

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Matrix multiplication The product of matrices A and B is deFned if the number of columns in A matches the number of rows in B . Defnition. Let A = ( a ik ) be an m × n matrix and B = ( b kj ) be an n × p matrix. The product AB is deFned to be the m × p matrix C = ( c ij ) such that c ij = n k =1 a ik b kj for all indices i , j . That is, matrices are multiplied row by column : p ∗ ∗ ∗ * * * P ∗ ∗ * ∗ ∗ * ∗ ∗ * = p ∗ ∗ ∗ ∗ ∗ ∗ * P
A = a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 . . . a mn = v 1 v 2 . . . v m B = b 11 b 12 . . . b 1 p b 21 b 22 . . . b 2 p . . . . . . . . . . . . b n 1 b n 2 . . . b np = ( w 1 , w 2 , . . . , w p ) = AB = v 1 · w 1 v 1 · w 2 . . . v 1 · w p v 2 · w 1 v 2 · w 2 . . . v 2 · w p . . . .

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