Solution_3 - THE CHINESE UNIVERSITY OF HONG KONG Department...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications (Fall 2007) Suggested Solutions for Homework 3 Solution 1. (a) This is a vector space. We must check all ten properties. i. ( α ) and ( β ); ii. A 1- A 4 ; iii. M 1- M 4 . ( α ) If x and y are positive reals, so is x + y = xy ( A 1 ) x + y = xy = yx = y + x . ( A 2 ) x + ( y + z ) = x ( yz ) = ( xy ) z = ( x + y ) + z . ( A 3 ) There is an object 0, the positive real number 1, which is such that 1 + x = 1 x = x = x 1 = x + 1 for all positive real number x . ( A 4 ) For all real x , the positive real 1 x acts as a negative: x + 1 x = x 1 x ¶ = 1 = 0 = 1 = 1 x ¶ x = 1 x + x. ( β ) If k is a real and x is a positive real, then kx = x k is again a positive real. ( M 1 ) k ( x + y ) = ( xy ) k = x k y k = kx + ky . ( M 2 ) ( k + l ) x = x k + l = x k x l = x k + x l . ( M 3 ) k ( lx ) = ( lx ) k = ( x l ) k = x lk = x kl = ( kl ) x . ( M 4 ) 1 x = x 1 = x . (b) This is a vector space. We must check all ten properties. i. ( α ) and ( β ); ii. A 1- A 4 ; iii. M 1- M 4 . ( α ) If we add two matrices of this form, the result will again be a matrix of this form • a b ‚ + • c d ‚ = • a + c b + d ‚ . ( A 1 ) • a b ‚ + • c d ‚ = • a + c b + d ‚ = • c + a d + b ‚ = • c d ‚ + • a b ‚ . 2 ( A 2 ) • a b ‚ + ˆ • c d ‚ + • e f ‚ ! = • a + ( c + e ) b + ( d + f ) ‚ = • ( a + c ) + e 0 ( b + d ) + f ‚ = ˆ • a b ‚ + • c d ‚ ! + • e f ‚ . ( A 3 ) The 2 × 2 zero matrix is of the appropriate form and has the desired properties. That is, • a b ‚ + • 0 0 0 0 ‚ = • a + 0 b + 0 ‚ = • a b ‚ = • 0 + a 0 0 + b ‚ = • 0 0 0 0 ‚ + • a b ‚ . ( A 4 ) If u is a matrix of the given form, then- u = •- a- b ‚ is again of the desired form and u + (- u ) = (- u ) + u = 0 ....
View Full Document

This note was uploaded on 05/18/2010 for the course MATHEMATIC MAT2310 taught by Professor Dr.jeffchak-fuwong during the Spring '06 term at CUHK.

Page1 / 6

Solution_3 - THE CHINESE UNIVERSITY OF HONG KONG Department...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online