TUTORIAL INTRODUCTION TO LINEAR ALGEBRA
Linear Algebra should be an essential tool for any engineer. Why? Because linear algebra has many application found in the behavior,
natural, physical and social sciences, in engineering, in business, in computer science and in pure and applied mathematics.
A simple application of linear algebra is finding the solution of a system
of linear equations in several unknowns.
I’ve produced a couple of videos involving linear algebra. Here is a simple linear algebra tutorial on multiplying matrices using Matlab.
Linear Algebra Tutorial Videos from the Khan Academy
Fortunately, the Khan Academy, a non-profit educational provider, offers over a hundred videos in linear algebra.
Below are the videos from the Khan Academy:
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The above set of linear algebra tutorial videos should give you a very good idea on what linear algebra is. Once you are comfortable with the above concepts in linear algebra you can proceed to more advanced applications on its use. Some of the key concepts given by these videos are: vectors, matrices, vector field, vector space and vector subspaces, eigenvalues, eigenvectors, linear transformations, singular matrices, and other linear algebra concepts.
Important concepts of linear algebra are vector spaces and linear maps between them. In more formal terms, a vector space is a set whose elements can be added together and multiplied by the scalars, or numbers.
In many physical applications, the scalars are real numbers. In general, the scalars may form any field F—thus one can consider vector spaces over the field Q of rational numbers, the field C of complex numbers, or a finite field Fq.
We’ll see in future courses in digital communications that the concept of vector spaces have been used in coding theory to correct errors in a digital message.
Vector operations of addition and scalar multiplication will have similar and usual addition and multiplication of numbers: addition is commutative and associative, multiplication distributes over addition, and so on.
Besides the mathematical formalism you need not to get to hung up on the technical details because there are many other applications in using linear algebra.
LINEAR ALGEBRA AND MODERN CONTROL THEORY
For example, Linear algebra concepts have been used in modern control systems. In modern control theory, you have matices A, B, C, D. Used in the following equation
x’=Ax+Bu (where x’ denotes first derivative of the state vector x)
where A is the system matrix. When finding the eigenvalues of the system matrix A, you have basically the system poles or it has a physical meaning of vibrational modes of a system such as a building or bridge. the vector x is the state vector. You can think of the displacements at various locations of a bridge as its system state. The matrix B, is the input distribution of the system. Physically, this is how the actuators of a system affect or influence the system by the input u. The matrix C, describes how sensors are distributed in the system and relates the state vector with the output vector y. The matrix D, decsribes how the input u directly affects the output y.
Based on the above description of a system, you can apply a variety of feedback schemes to improve system performance. For example, you can feed back either the state vector x or the output vector y to alter the states and feedback of a system to more desirable ones.
However, the scope of feedback and modern control theory is beyond the scope of this short blog article but hopefully you get an idea of what you can do with linear algebra.
If you are not in the United States, please come visit and replace 90% of our linear algebra professors. I am confident that these 7 minutes of your lecturing make more sense than an entire semester under their instruction. …
Publish Date: 08/30/2011 0:45
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