Lect24 - Chapter 4. Linear Transformations Math1111 Matrix...

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Unformatted text preview: Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 2 (Cont’d) Concrete example: Consider the linear functional L : R 3 → R defined as L (( x y z ) T ) = x + y + z . Find v ∈ R 3 so that R 3 = Span ( v ) ⊕ ker ( L ) . Give a geometric interpretation. Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 2 (Cont’d) Concrete example: Consider the linear functional L : R 3 → R defined as L (( x y z ) T ) = x + y + z . Find v ∈ R 3 so that R 3 = Span ( v ) ⊕ ker ( L ) . Give a geometric interpretation. Ans . A choice of v is ( 1 1 1 ) T . Simple calculation gives ker ( L ) = Span ( u 1 , u 2 ) where u 1 = ( 1- 1 ) T , u 2 = ( 1- 1 ) T . Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 2 (Cont’d) Concrete example: Consider the linear functional L : R 3 → R defined as L (( x y z ) T ) = x + y + z . Find v ∈ R 3 so that R 3 = Span ( v ) ⊕ ker ( L ) . Give a geometric interpretation. Ans . A choice of v is ( 1 1 1 ) T . Simple calculation gives ker ( L ) = Span ( u 1 , u 2 ) where u 1 = ( 1- 1 ) T , u 2 = ( 1- 1 ) T . W = Span ( v ) ker ( L ) = Span ( u 1 , u 2 ) Then, R 3 = W ⊕ ker ( L ) . Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 3 Example 3 . Let L : V → V be a linear operator, and W be a subspace of V . If W is invariant under L (i.e. L ( W ) ⊂ W ), show that one of its matrix representation of L is of the form where r = dim W . Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 3 Example 3 . Let L : V → V be a linear operator, and W be a subspace of V . If W is invariant under L (i.e. L ( W ) ⊂ W ), show that one of its matrix representation of L is of the form where r = dim W . Proof . Let w 1 , ··· , w r form a basis for W . Use Theorem 3.4.4 (ii) to extend these r linearly independent vectors to a basis for V , say, { w 1 , ··· , w r , x r + 1 , ··· , x n } . Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 3 Example 3 . Let L : V → V be a linear operator, and W be a subspace of V . If W is invariant under L (i.e. L ( W ) ⊂ W ), show that one of its matrix representation of L is of the form where r = dim W . Proof . Let w 1 , ··· , w r form a basis for W . Use Theorem 3.4.4 (ii) to extend these r linearly independent vectors to a basis for V , say, { w 1 , ··· , w r , x r + 1 , ··· , x n } . Then B = [ w 1 , ··· , w r , x r + 1 , ··· , x n ] is an ordered basis for V . The matrix representation of L w.r.t. B is of the desired form. Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 3 Example 3 . Let L : V → V be a linear operator, and W be a subspace of V . If W is invariant under L (i.e. L ( W ) ⊂ W ), show that one of its matrix representation of L is of the form where r = dim W ....
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Lect24 - Chapter 4. Linear Transformations Math1111 Matrix...

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