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Unformatted text preview: Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases  Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A = 1 . . . 1 . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases  Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A = 1 . . . 1 . Question How many 1 ’s are there? Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases  Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A = 1 . . . 1 . Question How many 1 ’s are there? Number of 1’s = rank ( A ) = dim L ( V ) . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases  Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A = 1 . . . 1 . Key : Let M be an m × n matrix. We can find suitable elementary matrices E 1 , ··· , E k , F 1 , ··· , F l so that E 1 ··· E k M F 1 ··· F l = 1 . . . 1 Call Q 1 = E 1 ··· E k & P = F 1 ··· F l . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases  Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A = 1 . . . 1 . Proof . Choose arbitrarily two ordered bases B 1 and B 2 . Let M = matrix representation r.t. B 1 and B 2 . By the key , there are P and Q such that Q 1 M P = A . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases  Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A = 1 . . . 1 . Proof . Choose arbitrarily two ordered bases B 1 and B 2 . Let M = matrix representation r.t. B 1 and B 2 ....
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 Spring '11

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