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# Lecture5 - Lecture 5 5.1 Linear Transformation(Conti Thm...

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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 0 ) ( x T solution set = ker(T) Nonhomogeneous equation b x T ) ( Solution set = { h h p ker(T)} if b p T ) ( for some
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Lecture5 - Lecture 5 5.1 Linear Transformation(Conti Thm...

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