Let f : R → R be a function with the following property: f(x + y) = f(x) + f(y) for all x, Y E R. Show that f(0) = 0. Show that f(-x) = -f(x). Show that f(x – y) = f(x) – f(y). 1. 2. 3. 4. Show: If n E N, then f(nx) = nf(x) and f(÷)= f(x) for all æ. Show: If r E Q, then f(rx) = rf(æ) for all x. 6. Show: If f is continuous at xo = 0, then f is continuous. 7. Show: If f is continuous, then there = cx. (Hint. exists c E R such that c = f(1).)

Let f : R → R be a function with the following property: f(x + y) = f(x) + f(y) for all x, Y E R. Show that f(0) = 0. Show that f(-x) = -f(x). Show that f(x – y) = f(x) – f(y). 1. 2. 3. 4. Show: If n E N, then f(nx) = nf(x) and f(÷)= f(x) for all æ. Show: If r E Q, then f(rx) = rf(æ) for all x. 6. Show: If f is continuous at xo = 0, then f is continuous. 7. Show: If f is continuous, then there = cx. (Hint. exists c E R such that c = f(1).)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 30EQ

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