2.4 Invertibility 2

2.4 Invertibility 2 - 2.4(2 R = T 1 Since T is invertible...

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: 2.4. (2) R = T- 1 Since T is invertible, ∃ T- 1 : V → V ⇒ T- 1 = T- 1 1 V = T- 1 ( TR ) = ( T- 1 T ) R = IR = R ∴ R = T- 1 11. T : P 3 ( R ) → M 2 × 2 ( R ) is linear by T ( f ) = f (1) f (2) f (3) f (4) ¶ We are going to show that T ( f ) = 0 ⇒ f = 0(the zero polynomial) In this case, f (1) = 0( i.e.f ( c ) = b ) ,f (2) = 0 ,f (3) = 0 ,f (4) = 0 ∴ ∀ f ( c i ) = 0 ,i = 0 , 1 , 2 , 3 ∴ f ( x ) = 3 ∑ i =0 b i f i ( x ) = 3 ∑ i =0 f ( c i ) f i ( x ) = 0 ∴ f is the zero polynomial ∴ T is one-to-one 12. φ β ( v i ) = [ v i ] β = e i = (0 , ··· , 1 , ··· , 0) t [ φ β ] γ β = I n , when γ is the standard basis for F n ∴ φ β is an isomorphism or φ β : V → F n is onto ⇒ φ β is an isomorphism because dimV = dimF n 13. ∼ is an equivalence relation on the class of vector space over F (i) ∼ is reflexive 107 PNU-MATH 2.4. ∀ V ∈ C ,V ∼ V ( ∵ ) I V : V → V s.t I V = v, ∀ v ∈ V is an isomorphism (ii) ∼ is symmetric If V ∼ W , then W ∼ V ( ∵ ) If T : V → W is isomorphic then ∃ T- 1 : W → V is isomorphic ∴ W ∼ V (iii) ∼ is transitive If V ∼ W and W ∼ Z , then V ∼ Z ( ∵ ) Let T : V → W and U : W → Z are isomorphic, then UT is isomorphic 14. Let V = { a a + b c ¶ | a,b,c ∈ F } T : V → F 3 s.t T a a + b c ¶ = ( a,b,c ) For the basis for V, { v 1 = 1 1 0 0 ¶ ,v 2 = 0 1 0 0 ¶ ,v 3 = 0 0 0 1 ¶ } ∃ ! T ( v i ) = w i is linear, w i ∈ F 3 i = 1 , 2 , 3 and since dimV = dimF 3 , V is isomorphic to F 3 ∴ T is an isomorphism from...
View Full Document

Page1 / 9

2.4 Invertibility 2 - 2.4(2 R = T 1 Since T is invertible...

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