Lect17

# Lect17 - Chapter 3 Vector Spaces Math1111 Change of Basis...

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Chapter 3. Vector Spaces Math1111 Change of Basis Change Coordinates - Example (Cont’d) Remark. Transition matrix in the last example is unique. i.e. if A is a matrix sending coodinate vectors w.r.t. [ u 1 , u 2 ] to coordinate vectors w.r.t. [ e 1 , e 2 ] , then A = 3 1 2 1 . The reason is that the coordinate of every vector is unique. Let A = a b c d . The coordinate vector of u 1 w.r.t. [ u 1 , u 2 ] is 1 0 . So the coordinate of u 1 w.r.t. [ u 1 , u 2 ] equals a c and also 3 2 . By uniqueness of the coordinate, a c = 3 2 .

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Chapter 3. Vector Spaces Math1111 Change of Basis Change Coordinates - Example (Cont’d) Example . Let { e 1 , e 2 } be the standard basis for R 2 , and let u 1 = ( 3 2 ) T , u 2 = ( 1 1 ) T . (ii) Find the coordinate vector w.r.t. [ u 1 , u 2 ] if its coordinate vector w.r.t. [ e 1 , e 2 ] is ( a b ) T .
Chapter 3. Vector Spaces Math1111 Change of Basis Change Coordinates - Example (Cont’d) Example . Let { e 1 , e 2 } be the standard basis for R 2 , and let u 1 = ( 3 2 ) T , u 2 = ( 1 1 ) T . (ii) Find the coordinate vector w.r.t. [ u 1 , u 2 ] if its coordinate vector w.r.t. [ e 1 , e 2 ] is ( a b ) T . Ans . Let ( α β ) T be the coordinate vector w.r.t. [ u 1 , u 2 ] . By Part (i), we have a b = U α β where U = 3 1 2 1 .

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Chapter 3. Vector Spaces Math1111 Change of Basis Change Coordinates - Example (Cont’d) Example . Let { e 1 , e 2 } be the standard basis for R 2 , and let u 1 = ( 3 2 ) T , u 2 = ( 1 1 ) T . (ii) Find the coordinate vector w.r.t. [ u 1 , u 2 ] if its coordinate vector w.r.t. [ e 1 , e 2 ] is ( a b ) T . Ans . Let ( α β ) T be the coordinate vector w.r.t. [ u 1 , u 2 ] . By Part (i), we have a b = U α β where U = 3 1 2 1 . Hence, α β = U - 1 a b .
Chapter 3. Vector Spaces Math1111 Change of Basis Change Coordinates - Example (Cont’d) Example . Let { e 1 , e 2 } be the standard basis for R 2 , and let u 1 = ( 3 2 ) T , u 2 = ( 1 1 ) T . (ii) Find the coordinate vector w.r.t. [ u 1 , u 2 ] if its coordinate vector w.r.t. [ e 1 , e 2 ] is ( a b ) T . Ans . Let ( α β ) T be the coordinate vector w.r.t. [ u 1 , u 2 ] . By Part (i), we have a b = U α β where U = 3 1 2 1 . Remark. A transition matrix is always nonsingular. (Why?) Two ways: 1) For what coordinate vector is it sent to the zero vector? 2) Use the result in Further Example 2.

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Chapter 3. Vector Spaces Math1111 Vector Spaces Homework 10 Reading Leon (7th edition): p.151 - p.157 Leon (8th edition): p.144 - p.149 Homework 10 Leon (7th edition): Chapter 3 - Section 5 Qn. 1-3, 5. Leon (8th edition): Section 3.5 Qn. 1-3, 5.
Chapter 3. Vector Spaces Math1111 Change of Basis Change Coordinates - Further Example 1 Example . Let E = [ e 1 , ··· , e n ] be the standard ordered basis for R n . Suppose b 1 , ··· , b n R n also form a basis for R n . Let F = [ b 1 , ··· , b n ] .

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