Lect18 - Chapter 3. Vector Spaces Math1111 Change of Basis...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Change of Basis Change Basis for General Vector Spaces (Cont’d) Definition Let E = [ v 1 , v 2 , ··· , v n ] and F = [ w 1 , w 2 , ··· , w n ] be ordered bases for V . The transition matrix from E to F is the matrix S satisfying [ v ] F = S [ v ] E for any v ∈ V . Chapter 3. Vector Spaces Math1111 Change of Basis Change Basis for General Vector Spaces (Cont’d) Definition Let E = [ v 1 , v 2 , ··· , v n ] and F = [ w 1 , w 2 , ··· , w n ] be ordered bases for V . The transition matrix from E to F is the matrix S satisfying [ v ] F = S [ v ] E for any v ∈ V . Chapter 3. Vector Spaces Math1111 Change of Basis Change Basis for General Vector Spaces - Example Example . Find the coordinate vectors of a b c d ∈ R 2 × 2 w.r.t. (i) E = [ e 1 , e 2 , e 3 , e 4 ] where e 1 = 1 , e 2 = 1 , e 3 = 1 , e 4 = 1 . (ii) F = [ u 1 , u 2 , u 3 , u 4 ] where u 1 = 1 1 , u 2 = 1 1 , u 3 = - 1 1 , u 4 = 1- 1 . Find the transition matrix from F to E . Chapter 3. Vector Spaces Math1111 Change of Basis Change Basis - Example (Cont’d) Ans . Let A = a b c d ∈ R 2 × 2 . (i) The coordinate vector of A w.r.t. E , [ A ] E = ( a b c d ) T . Chapter 3. Vector Spaces Math1111 Change of Basis Change Basis - Example (Cont’d) Ans . Let A = a b c d ∈ R 2 × 2 . (i) The coordinate vector of A w.r.t. E , [ A ] E = ( a b c d ) T . Because A = a 1 + b 1 + c 1 + d 1 Chapter 3. Vector Spaces Math1111 Change of Basis Change Basis - Example (Cont’d) Ans . Let A = a b c d ∈ R 2 × 2 . (i) The coordinate vector of A w.r.t. E , [ A ] E = ( a b c d ) T . (ii) The coordinate vector of A w.r.t. F , [ A ] F = ( d + a 2 b + c 2 d- a 2 b- c 2 ) T . Because A = d + a 2 1 1 + b + c 2 1 1 + d- a 2 - 1 1 + b- c 2 1- 1 Chapter 3. Vector Spaces Math1111 Change of Basis Change Basis - Example (Cont’d) Ans . Let A = a b c d ∈ R 2 × 2 . (i) The coordinate vector of A w.r.t. E , [ A ] E = ( a b c d ) T . (ii) The coordinate vector of A w.r.t. F , [ A ] F = ( d + a 2 b + c 2 d- a 2 b- c 2 ) T . The transition matrix from F to E is 1- 1 1 1 1- 1 1 1 . Chapter 3. Vector Spaces Math1111 Change of Basis Change Basis - Example (Cont’d) Ans . Let A = a b c d ∈ R 2 × 2 ....
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Lect18 - Chapter 3. Vector Spaces Math1111 Change of Basis...

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