{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

10aAss4

# 10aAss4 - AS4/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG...

This preview shows pages 1–2. Sign up to view the full content.

AS4/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 4 Due date : Nov 25, 2010 before 6:30 p.m. Part I: Hand-in your solutions 1. Let a = 1 3 ; b = 2 2 ; c = 3 1 be three vectors in R 2 . (a) Is it possible to nd a linear transformation L : R 2 ! R 2 such that L ( a ) = b ; L ( b ) = c ; L ( c ) = a ? Explain your answer. (b) De ne the linear transformation T : R 2 ! R 2 by T ( x y ) = 8 x + 6 y 9 x + 7 y : i. Evaluate the matrix representation A of T relative to the ordered bases E = [ a ; b ] and F = [ b ; c ], i.e. [ T ( x )] F = A [ x ] E for all x 2 R 2 . ii. Find an ordered basis G of R 2 such that the matrix representation of T with respect to G (i.e. relative to G and G ) is a diagonal matrix. 2. (a) Let A = 0 @ 1 1 2 0 1 1 1 3 4 1 A . i. Show that R 3 = N ( A ) c ( A ), where N ( A ) is the nullspace and c ( A ) is the column space of A . ii. Is it always true that R 3 = N ( B ) + c ( B ) for any 3 3 matrix B ? (b) Let L : V ! V be a linear operator. If L L = L , i.e. L ( v ) = L ( L ( v )), show that V = ker( L ) L ( V ).

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.

{[ snackBarMessage ]}

### Page1 / 3

10aAss4 - AS4/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online