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1: Let V be a finite dimensional vector space and W1, W2 be non-trivial subspaces of V such that W1 n W2 = {0v}. We say that V is the direct sum of W1 and W2 if any v e V, v = a + b where a e W1 and b e W2. In this case we write V = W1 W2. Let T : V V be a linear transformation such that T2 = T. Show that V = ker T Range T.

1: Let V be a finite dimensional vector space and W1, W2 be non-trivial subspaces of V such that W1 n W2 = {0v}. We say that V is the direct sum of W1 and W2 if any v e V, v = a + b where a e W1 and b e W2. In this case we write V = W1 W2. Let T : V V be a linear transformation such that T2 = T. Show that V = ker T Range T.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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1. kerT ∩ RangeT

2. v = a + b for v ∈ V, a ∈ kerT and b ∈ RangeT

1: Let V be a finite dimensional vector space and W1, W2 be non-trivial subspaces of V such that W1 n W2 = {0v}. We say that V is the direct sum of W1 and W2 if any v e V, v = a + b where a e W1 and b e W2. In this case we write V = W1 e W2. Let T :V → V be a linear transformation such that T2 = T. Show that | V = ker T Range T. %3D
Transcribed Image Text:1: Let V be a finite dimensional vector space and W1, W2 be non-trivial subspaces of V such that W1 n W2 = {0v}. We say that V is the direct sum of W1 and W2 if any v e V, v = a + b where a e W1 and b e W2. In this case we write V = W1 e W2. Let T :V → V be a linear transformation such that T2 = T. Show that | V = ker T Range T. %3D
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