Linear Algebra: Systems and Operations
Dive into the world of Linear Algebra to explore linear systems, operations like multiplication and addition, vectors, transformations, and more. Enhance your knowledge of linear systems with examples and explanations, and discover the intricacies of linear functions and derivatives. Explore the concept of linearity in various mathematical contexts and understand the importance of matrices in solving problems efficiently.
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Presentation Transcript
What to learn in Linear Algebra? Linear System
A system has input and output (function, transformation, operator) System Speech Recognition System How are you Dialogue System (e.g. Siri, Alexa) How are you I am fine Communication System Hello Hello
Linear System 1. Persevering Multiplication Linear System Linear System ? ? 2. Persevering Addition Linear System + + Linear System Linear System
Linear System When the input and output are vectors 1. Persevering Multiplication 3 4 5 3 4 5 Linear System 1 2 ?1 Linear System ? 2 ? = 2 6 8 10 2 4 2. Persevering Addition 3 4 5 7 8 9 7 8 9 3 4 5 1 2 Linear System + 1 2+5 6 Linear System 5 6 Linear System 10 12 14 6 8
Linear? NO x2 x System 1. Persevering Multiplication k2x2 kx2 x2 x kx System System 2. Persevering Addition ?1 2 ?1 System 2 ?1+ ?2 ?1+ ?2 System 2 ?2 ?2 System 2+ ?2 2 ?1
6 8 9 9 0 2 Linear? 6 9 8 0 9 2 System Transpose ? ? YES 6? 8? 9? 9? 0 2? 6? 9? 8? 0 9? 2? System 6 8 9 9 0 2 System 6 9 8 0 9 2 7 13 5 8 10 12 System 1 2 3 4 5 6 System 7 13 10 5 12 8 1 4 2 5 3 6
Linear? YES function f function f Derivative ? ? ? ? x2 2x ?? ?? ? + ? ? + ? 3 3x
Linear? 1 3?3 ?3 ? ? ? = ?2 Integral (from a to b) ? ? ?? Function ? ? ? scalar Area a b
? ? ? ?? ? ? Linear? YES ? ? Persevering Multiplication ?? ? ?? ? ?? ? ? = ? ? ? ?? ? ? ? Persevering Addition ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ? + ? ? ?? ? ? + ? ? ? ? ? = ? ? ?? + ? ? ?? ? ?
