Linear Algebra Done Right

Abstract

Undergraduate Texts in Mathematics are generally aimed at third-and fourthyear undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an appreciation of interrelations among different aspects of the subject. They feature examples that illustrate key concepts as well as exercises that strengthen understanding.

Mathematical Inequalities

“…We first establish the following definitions and lemmas [Axl97]. Otherwise, the set is linearly dependent.…”

“…Because ሺ , … , ሻ is orthonormal, we have ‫ۃ‬ , ‫ۄ‬ ൌ 0 for ݅ ് ݆ and ‫ۃ‬ , ‫ۄ‬ ൌ 1 for ݅ ൌ ݆ (where Proof: See Axler [Axl97].…”

Energy efficient error-correcting coding for wireless systems

“…• Only when the dynamic range of the channel D changes from (10,20] dB to (20,30] dB, FEC II loses around 2 dB in SNR to achieve the error-free quality. From D ≥ 20 dB, there is no performance loss as D increases.…”

“…This new scheme works better in practice than the joint coding scheme over the wireless channel at any D. With respect to the LDPC code from the IEEE 802.11n standard, on average, this novel approach gains a SNR of around 1.5 dB at D ∈ (0,10] dB, around 1.7 dB at D ∈ (10,20] dB and around 3.8 dB at D ∈ (20,30] dB. Besides, opportunistic error correction can offer us the error-free quality but the LDPC code defined in the IEEE 802.11n standard can only achieve a BER of 10 −4 at D ∈ (0,10] dB, 1.4 × 10 −4 at D ∈ (10,20] dB and 3 × 10 −4 at D ∈ (20,30] dB. Their performance difference becomes larger as D increases, as shown in the simulation and in the practical measurement.…”

“…More measurements succeed in opportunistic error correction than in the joint coding scheme. This new scheme works better in practice than the joint coding scheme over the wireless channel at any D. With respect to the joint coding scheme, on average, this novel approach gains a SNR of around 1.5 dB at D ∈ (0,10] dB, around 1.7 dB at D ∈ (10,20] dB and around 3.8 dB at D ∈ (20,30] dB.…”