Introduction to Linear Equations | Linear Algebra #6

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The sixth lecture of the "Linear Algebra" series is entitled "Introduction to Linear Equations".

A system of n linear equations in n unknowns x1, x2, . . . , xn is a family of equations
a11x1 +a12x2 +···+a1nxn = b1
a21x1 +a22x2 +···+a2nxn = b2
. . .
an1x1 +an2x2 +···+annxn = bn

We wish to determine if such a system has a solution, that is to find out if there exist numbers x1 , x2 , . . . , xn that satisfy each of the equations simultaneously. We say that the system is consistent if it has a solution. Otherwise, the system is called inconsistent.

Geometrically, solving a system of linear equations in two (or three) unknowns is equivalent to determining whether or not a family of lines (or planes) has a common point of intersection.

The solution of linear systems of equations is of primary importance in linear algebra. The problem of solving a linear system arises in almost all areas of engineering and science, including the structure of materials, statics and dynamics, the design and analysis of circuits, quantum physics, and computer graphics. The solution to linear systems also hides under the surface in many methods. For instance, a standard tool for data fitting is cubic splines. The fit is found by finding the solution to a system of linear equations.

00:00 Applications of Linear Equations
02:13 What are Linear Equations ?
04:18 System of Linear Equations
07:57 Polynomial Fitting and Interpolation
11:19 Summary

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[kaylahreichert]

best linear algebra series I've seen so far. I've watched them all so far.

[daveframi]

Great. Can't wait for this to premiere !!

[peterkelly]

bravo MR. Ahmad Bazzi, keep up the good work, but please don't forget your Convex Optimization series? What happened that you stopped all of a sudden ?

[reyeshyatt]

Is there a website like you can write down all the equations ?? thanks

[gladyssigler]

do a lesson on Expansion by Minors to compute determinants. Thanks !!

[aaronsmart]

please I urge you to talk about basis of a subspace. See, I'm working on my PhD to find a numerical way to orthonormalize a basis of any given set of vectors in a time efficient manner. I know that the Gram-Schmidt algorithm for computing an orthonormal basis is time-honored and important. Also, the conjugate gradient method for the solution of large, sparse symmetric positive definite systems is presented. This method is one of the essences of numerical linear algebra and has revolutionized the solution of many very large problems. When can you touch upon this area ?

[lourdesgottlieb]

Do you think you could show us a numerical application ? I know that this easily applies to electrical circuits involving RLC. Thanks.

[louisemcleod]

great initiative to start a new series on linear algebra, do you think you could also do a series on microprocessors. I'm an Electrical Engineering student and I'm having a hard time working with my STM32 Microcontroller. IN precise, it is the ARM Cortex M4 based STM32F407 DISCOVERY board from ST. Do you think you could some embedded C lectures for microprocessor firmware? thanks.

[verliejaeger]

But wait, if this is the linear systems, when do you teach how to solve them ?

[dieknappegast]

how tf did i get here pop up links are really anoying but good vid tho