Understanding Vector Operations in Linear Algebra
Explore the world of vector operations in linear algebra through this detailed presentation. Learn about vector addition, scalar multiplication, field operations, and more. Gain insights into the notation of Fn and the significance of scalar multiplication and vector addition in linear algebra. Whether dealing with real or complex vectors, grasp the essential concepts for further studies in this field.
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NE 112 Linear algebra for nanotechnology engineering 2.3 Vector operators Douglas Wilhelm Harder, LEL, M.Math. dwharder@uwaterloo.ca dwharder@gmail.com
Vector operations Introduction In this topic, we will Revisit field operations Introduce vector operations Describe the notation of Fn 2
Vector operations Vector operations In a field, we had two operations: Addition Multiplication For vectors, we will introduce two operations: Scalar multiplication Vector addition Important: while we could define vector multiplication, there are no significant consequences that can be deduced The properties of linear algebra depend on scalar multiplication and vector addition only 3
Vector operations Vector operations We have looked at real and complex vectors Both the real numbers and the complex numbers are fields If we don t care if we are using real or complex vectors, we will use the notation Fn where it is understood that the field F can be either R or C The entries of the corresponding field are also called scalars We will be: Adding two vectors Multiplying a vector by a scalar or entry in the corresponding field 4
Vector operations Summary Following this topic, you now Have been introduced to the operations we will focus on next Vector addition and scalar multiplication Know we will use Fnwhen we don t care if the vectors are real or complex Understand that scalar is just another term for an entry in the corresponding field 5
Vector operations References [1] https://en.wikipedia.org/wiki/Linear_algebra 6
Vector operations Acknowledgments None so far. 7
Vector operations Colophon These slides were prepared using the Cambria typeface. Mathematical equations use Times New Roman, and source code is presented using Consolas. The photographs of flowers and a monarch butter appearing on the title slide and accenting the top of each other slide were taken at the Royal Botanical Gardens in October of 2017 by Douglas Wilhelm Harder. Please see https://www.rbg.ca/ for more information. 8
Vector operations Disclaimer These slides are provided for the NE 112 Linear algebra for nanotechnology engineering course taught at the University of Waterloo. The material in it reflects the authors best judgment in light of the information available to them at the time of preparation. Any reliance on these course slides by any party for any other purpose are the responsibility of such parties. The authors accept no responsibility for damages, if any, suffered by any party as a result of decisions made or actions based on these course slides for any other purpose than that for which it was intended. 9
