Lecture5 - Lecture 5 5.1 Linear Transformation (Conti.)...

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1 Lecture 5 5.1 Linear Transformation (Conti.) Thm: Preservation of subspaces Let V V , be vector spaces and V V T : be a linear transformation. 1. If W is a subspace of V, then T[W] is a subspace of V . 2. If W´ is a subspace of V , then T -1 [W] is a subspace of V. 5.2 Kernel and image of a linear transformation Let V V , be vector spaces and V V T : be a linear transformation. ker(T) =   V v v T v , 0 ) ( Range(T) =   V v v T ) ( Example: If V V T : is a linear transformation, show that Ker(T) is a subspace of V and range(T) is a subspace of V .
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2 Example: Define a transformation T: M nn → M nn by T(A) = A – A T for all A in M nn . Show that T is linear and that a) ker(T) consists of all symmetric matrices; b) range(T) consists of all skew-symmetric matrices. Let V V , be vector spaces and V V T : be a linear transformation. Homogeneous equation
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This note was uploaded on 12/25/2010 for the course MATHEMATIC MATB24 taught by Professor Yang during the Fall '09 term at University of Toronto- Toronto.

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Lecture5 - Lecture 5 5.1 Linear Transformation (Conti.)...

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