{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

5.1 (2,3,4,10,11,14,15,17,19,22) 5.2(2,3,8,9,10,13)

# 5.1 (2,3,4,10,11,14,15,17,19,22) 5.2(2,3,8,9,10,13) -...

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

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: Selected answers to suggested problems: 5.1, 5.2 5.1 2. (b) T (3 + 4 x ) =- 2(3 + 4 x ), T (2 + 3 x ) =- 3(2 + 3 x ), so both are eigenvectors (and a basis) and [ T ] β =- 2- 3 . (d) T ( x- x 2 ) =- 4(- 1- x + x 2 ), T (- 1 + x 2 ) =- 2(- 1 + x 2 ), T (- 1- x + x 2 ) = 3( x- x 2 ). This is not a basis of eigenvectors; [ T ] β = 3- 2- 4 . (f) T 1 1 =- 3 1 1 , T- 1 2 =- 1 2 , T 1 2 = 1 2 , and T- 1 2 =- 1 2 , so this is a basis of eigenvectors, and [ T ] β = - 3 1 1 1 . 3. (b) (i) eigenvalues 1, 2, 3 (ii) E 1 = - z- z z : z ∈ R . E 2 = x- x : x ∈ R . E 3 = x- x : x ∈ R . (iii) One possible basis is { (1 , 1 ,- 1) , (1 ,- 1 , 0) , (1 , ,- 1) } . (iv) Q = 1 1 1- 1- 1- 1- 1 . D = 1 2 3 . (d) (i) eigenvalues 0, 1, 1 1 2 (ii) E = 1 2 z- 6 z z : z ∈ R . E 1 = y : y ∈ R . (iii) Not possible; E 1 is not of large enough dimension. 4. (c) Matrix representation of T in terms of the standard ordered basis is - 4 3- 6 6- 7 12 6- 6 11 . Determinant of [ T ]- tI is 2 + 3 t- t 3 ; eigenvalues 2 ,- 1 ,- 1. E 2 = 1 2 z z z : z ∈ R . E- 1 = y + 2 z y z : y, z ∈ R . One possible basis: {- 1 2 , 1 , 1) , (1 , 1 , 0) , (2 , , 1) } . (d) Matrix representation of T in terms of standard ordered basis ( { 1 , x } ) is 1- 6 2- 6 . Determinant of [ T ]- tI is 6 + 5 t + t 2 ; eigenvalues- 2 ,- 3. E- 2 = 2 y y : y ∈ R . E- 3 = 3 a 2 a : a ∈ R . One possible basis: { 2 + x, 3 + 2 x } . (e) Matrix representation of T in terms of standard ordered basis ( { 1 , x, x 2 } ) is 1 3 9 1 3 4 2 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

5.1 (2,3,4,10,11,14,15,17,19,22) 5.2(2,3,8,9,10,13) -...

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

View Full Document
Ask a homework question - tutors are online