13. Recall from Section 2.1 that the set of positive real numbers R+ is a vector space under the "addi- tion" x y = xy and the "scalar multiplication" cox=xc. TOPSTO a) Show that the natural logarithm is a linear transformation from R+ to R. b) Show that the exponential function is a linear transformation from R to R+.
13. Recall from Section 2.1 that the set of positive real numbers R+ is a vector space under the "addi- tion" x y = xy and the "scalar multiplication" cox=xc. TOPSTO a) Show that the natural logarithm is a linear transformation from R+ to R. b) Show that the exponential function is a linear transformation from R to R+.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:**Section 13: Positive Real Numbers as a Vector Space**
Recall from Section 2.1 that the set of positive real numbers \(\mathbb{R}^+\) is a vector space under the operations:
- **"Addition"**: \(x \oplus y = xy\)
- **"Scalar Multiplication"**: \(c \odot x = x^c\)
**Tasks:**
a) Demonstrate that the natural logarithm is a linear transformation from \(\mathbb{R}^+\) to \(\mathbb{R}\).
b) Prove that the exponential function is a linear transformation from \(\mathbb{R}\) to \(\mathbb{R}^+\).
Expert Solution
Step 1
To answer the question we use the properties of natural logarithm and exponential functions and use the given definition of addition and scalar multiplication as follows.
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