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Read the following definition, and then answer the question that follows. Suppose that T : R" → R" is a function (not necessarily linear). The core of T, written core(T), is the set of x E R" so that T(x) = 0 (it is the set of vectors sent to 0 by the transformation.) 2.1 Show that if T : R" → R" is a linear transformation, and x, y e core(T), c e R, then x+y € core(T) and cx E core(T). 2.2 Show that if T : R" → R" is such that 0 is not in core(T), then T is not a linear transformation. Definition

Read the following definition, and then answer the question that follows. Suppose that T : R" → R" is a function (not necessarily linear). The core of T, written core(T), is the set of x E R" so that T(x) = 0 (it is the set of vectors sent to 0 by the transformation.) 2.1 Show that if T : R" → R" is a linear transformation, and x, y e core(T), c e R, then x+y € core(T) and cx E core(T). 2.2 Show that if T : R" → R" is such that 0 is not in core(T), then T is not a linear transformation. Definition

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.1: Introduction To Linear Transformations
Problem 39E: For the linear transformation from Exercise 33, find a T(1,1), b the preimage of (1,1), and c the...
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Read the following definition, and then answer the question that follows. Suppose that T : R" → R" is a function (not necessarily linear). The core of T, written core(T), is the set of x E R" so that T(x) = 0 (it is the set of vectors sent to 0 by the transformation.) 2.1 Show that if T : R" → R" is a linear transformation, and x, y e core(T), c e R, then x + y € core(T) and cx E core(T). 2.2 Show that if T : R" → R" is such that 0 is not in core(T), then T is not a linear transformation. Definition
Transcribed Image Text:Read the following definition, and then answer the question that follows. Suppose that T : R" → R" is a function (not necessarily linear). The core of T, written core(T), is the set of x E R" so that T(x) = 0 (it is the set of vectors sent to 0 by the transformation.) 2.1 Show that if T : R" → R" is a linear transformation, and x, y e core(T), c e R, then x + y € core(T) and cx E core(T). 2.2 Show that if T : R" → R" is such that 0 is not in core(T), then T is not a linear transformation. Definition
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