hw1_sol - Department of Electrical and Computer Engineering...

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

View Full Document Right Arrow Icon
Department of Electrical and Computer Engineering University of Maryland, College Park Fall 2011 ENEE 660 Homework Set 1 Solution Nuno C. Martins 1. Exercise 1.2 Solution. U = { x R 5 | [x] 1 = 3[x] 2 , [x] 3 = 7[x] 4 } Choose b 1 = 3 1 0 0 0 ,b 2 = 0 0 7 1 0 ,b 3 = 0 0 0 0 1 . First, observe that { b 1 ,b 2 ,b 3 } are linearly independent. Claim. span { b 1 ,b 2 ,b 3 } = U U span { b 1 ,b 2 ,b 3 } x U, [ x ] 1 = 3[ x ] 2 , [ x ] 3 = 7[ x ] 4 , thus, x = 3 [ x ] 2 [ x ] 2 7 [ x ] 4 [ x ] 4 [ x ] 5 = [ x ] 2 b 1 + [ x ] 4 b 2 + [ x ] 5 b 3 span { b 1 ,b 2 ,b 3 } span { b 1 ,b 2 ,b 3 } ⊆ U x span { b 1 ,b 2 ,b 3 } , α 1 2 3 s.t. x = α 1 b 1 + α 2 b 2 + α 3 b 3 = x = 3 α 1 α 1 7 α 2 α 2 α 3 U From above, span { b 1 ,b 2 ,b 3 } = U , and { b 1 ,b 2 ,b 3 } are linearly independent. Therefore, { b 1 ,b 2 ,b 3 } are one basis for U . (Ming Tse P. Laiu, 2011) 2. Exercise 1.3 Solution. By dimension theorem, dim ( U + W ) = dim ( U ) + dim ( W ) - dim ( U W ) dim ( U W ) = 0 Since U , W are subspaces, { 0 } ∈ U , { 0 } ∈ W ⇒ { 0 } ∈ U W U W = { 0 } (Ming Tse P. Laiu, 2011)
Background image of page 1

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

View Full DocumentRight Arrow Icon
3. Exercise 1.4 Solution. The statement is NOT true. Consider this counterexample below: U 1 ,U 2 ,U 3 R 2 U 1 = span { (0 , 1) } , U 2 = span { (1 , 0) } , U 3 = span { (1 , 1) } (Ming Tse P. Laiu, 2011) 4. Exercise 1.5 Solution. Denote ~x = ( x 1 ,x 2 ) ,~ y = ( y 1 ,y 2 ). From definition of triangle function, sin θ 1 = x 2 p x 2 1 + x 2 2 cos θ 1 = x 1 p x 2 1 + x 2 2 sin θ 2 = y 2 p y 2 1 + y 2 2 cos θ 2 = y 2 p y 2 1 + y 2 2 Therefore, cos θ = cos( θ 2 - θ 1 ) = cos θ 2 cos θ 1 + sin θ 2 sin θ 1 = x 2 y 2 + x 1 y 1 p x 2 1 + x 2 2 · p y 2 1 + y 2 2
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/26/2012 for the course ECE 660 taught by Professor Nunomartins during the Fall '11 term at Maryland.

Page1 / 8

hw1_sol - Department of Electrical and Computer Engineering...

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