Lect21 - Chapter 4. Linear Transformations Math1111 Linear...

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Chapter 4. Linear Transformations Math1111 Linear Transformations Kernel Definition Let L : V W be a linear transformation. The kernel of L , denoted by ker ( L ) , is defined as ker ( L ) : = { v V : L ( v ) = 0 } . Example . Find ker ( L ) where L : R 2 R 3 is defined by L (( a b ) T ) = ( b 0 a ) T . Example . Let A be an 3 × 2 matrix. Define L : R 2 R 3 by L ( x ) = A x for any x R 2 . Show that L is a linear transformation, and ker ( L ) = N ( A ) .
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Chapter 4. Linear Transformations Math1111 Linear Transformations Definition Let L : V W be a linear transformation, and S be subspace of V . The image of S under L , denoted by L ( S ) , is defined as L ( S ) : = { w W : w = L ( v ) for some v S } . We call L ( V ) the range of L .
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Chapter 4. Linear Transformations Math1111 Linear Transformations Definition Let L : V W be a linear transformation, and S be subspace of V . The image of S under L , denoted by L ( S ) , is defined as L ( S ) : = { w W : w = L ( v ) for some v S } . We call L ( V ) the range of L . Example . Find L ( S ) where L : R 2 R 3 is defined by L (( a b ) T ) = ( b 0 a ) T , and S = Span ( e 1 + e 2 ) R 2 .
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Chapter 4. Linear Transformations Math1111 Linear Transformations Basic Properties Theorem 4.1.1 Let L : V W be a linear transformation, and S be a subspace of V . Then (i) ker ( L ) is a subspace of V , (ii) L ( S ) is a subspace of W .
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Chapter 4. Linear Transformations Math1111 Linear Transformations Homework 1 Reading Leon (7th edition): p.175-182 Leon (8th edition): p.166-174 Homework 1 Leon (7th edition): Chapter 4 Section 1 Qn. 1, 2, 12-14, 17, 19,
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Lect21 - Chapter 4. Linear Transformations Math1111 Linear...

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