Solutions E Answers and Solutions to Homeworks
Part I Orthogonality
Week 1 Norms
Section 1.1 Opening Remarks
Unit 1.1.1 Why norms?
Homework 1.1.1.1.
Homework 1.1.1.2.
Homework 1.1.1.3.
Homework 1.1.1.4.
Homework 1.1.1.5.
Section 1.2 Vector Norms
Unit 1.2.1 Absolute value
Homework 1.2.1.1.
Homework 1.2.1.2.
Homework 1.2.1.3.
Homework 1.2.1.4.
Homework 1.2.1.5.
Homework 1.2.1.6.
Unit 1.2.2 What is a vector norm?
Homework 1.2.2.1.
Unit 1.2.3 The vector 2-norm (Euclidean length)
Homework 1.2.3.2.
Homework 1.2.3.3.
Unit 1.2.4 The vector \(p\)-norms
Homework 1.2.4.1.
Homework 1.2.4.2.
Unit 1.2.5 Unit ball
Homework 1.2.5.1.
Unit 1.2.6 Equivalence of vector norms
Homework 1.2.6.1.
Homework 1.2.6.2.
Homework 1.2.6.3.
HintSolution 1 \(\| x \|_1 \leq C_{1,2} \| x \|_2 \)Solution 2 \(\| x \|_1 \leq C_{1,\infty} \| x \|_\infty\)Solution 3 \(\| x \|_2 \leq C_{2,1} \| x \|_1 \text{:}\)Solution 4 \(\| x \|_2 \leq C_{2,\infty} \| x
\|_\infty \)Solution 5 \(\| x \|_\infty \leq C_{\infty,1} \| x \|_1 \text{:}\)Solution 6 \(\| x \|_\infty \leq C_{\infty,2} \| x \|_2\)Solution 7 Table of constants