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File: 1608528241467.png ( 28.54 KB , 677x492 , Linear_subspaces_with_shad….png )

-Linear Algebra General-

Welcome to /LA/ comrades. In this thread we will work together more or less in line with the MIT OCW Linear Algebra syllabus.

The OCW page can be found here: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/index.htm

On the OCW page you can find the calendar, recommended readings, lectures, and problem sets and exams. The lectures are done by Gilbert Strang who also wrote the recommended textbook. I think he is a very good instructor and I believe you should certainly give his lectures a watch if you are interested in learning more.

The Calendar is divided into 40 sessions which correspond to 40 assigned readings and lectures. There are 10 problem sets and 4 exams with all the solutions online. This thread will serve as a place to discuss lectures, readings, and, probably most usefully, ask other anons for help on problem sets or exams.

Welcome to /LA/ comrades. In this thread we will work together more or less in line with the MIT OCW Linear Algebra syllabus.

The OCW page can be found here: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/index.htm

On the OCW page you can find the calendar, recommended readings, lectures, and problem sets and exams. The lectures are done by Gilbert Strang who also wrote the recommended textbook. I think he is a very good instructor and I believe you should certainly give his lectures a watch if you are interested in learning more.

The Calendar is divided into 40 sessions which correspond to 40 assigned readings and lectures. There are 10 problem sets and 4 exams with all the solutions online. This thread will serve as a place to discuss lectures, readings, and, probably most usefully, ask other anons for help on problem sets or exams.

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-Useful Resources-

Instead of using Matlab (proprietary and therefore costly software for non students) you may choose to use GNU Octave, an open source (and free as in gratis) alternative.

https://www.gnu.org/software/octave/

The video lectures can be found here:

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

The problem sets can be found here:

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/assignments/

And the exams can be found here:

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/exams/

Instead of using Matlab (proprietary and therefore costly software for non students) you may choose to use GNU Octave, an open source (and free as in gratis) alternative.

https://www.gnu.org/software/octave/

The video lectures can be found here:

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

The problem sets can be found here:

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/assignments/

And the exams can be found here:

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/exams/

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Here's a question I have for the reading assigned for session 1. Strang introduces the Schwarz inequality which is the proposition that, "Whatever the angle, this dot product of v/||v|| with w/||w|| never exceeds one." Could other Anons explain the proof behind this? Specifically Proof 1 on the related wikipedia page ( https://en.wikipedia.org/wiki/Cauchy–Schwarz_inequality ) I'm a bit hung up on the notation. For instance what does u subscript v indicate?

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File: 1608528243706.png ( 17.09 KB , 468x174 , Screenshot 2020-08-26 at 0….png )

>>2967

So I will just go through the proof from basics.

We seek to prove that

|<u,v>| =< |u||v|

If we have either vector=0 then this holds trivially.

Now to prove the case where v is non-zero we set

z=u-u_v

Where u_v = <u,v>/<v,v> * v

It is then showed that z is such that it is orthogonal to v (by the property of dot products).

So we now have 3 vectors. u and v, which are any nontrivial vector, and z which is orthogonal to v. From our definition of z, we have from a simple rearrangement

u = u_v + z

Basically we can then square the norms of everything on both sides from the Pythagorean theorem which is pic related. Basically we technically have a sum on both sides. So we take the norm and square it (left hand side of the equation), then from the right hand side (usually we call it RHS) we can just expand it out.

Using various properties of dot products we can rearrange this to get Cauchy-Schwarz. Do let me know if you'd like me to clarify any of this. :)

So we can see that u subscript v is just a vector that is useful for us to get u in a form that is nice to play with. In a more intuitive sense u_v is the magnitude of the dot product between u and v, in the direction of v.

So I will just go through the proof from basics.

We seek to prove that

|<u,v>| =< |u||v|

If we have either vector=0 then this holds trivially.

Now to prove the case where v is non-zero we set

z=u-u_v

Where u_v = <u,v>/<v,v> * v

It is then showed that z is such that it is orthogonal to v (by the property of dot products).

So we now have 3 vectors. u and v, which are any nontrivial vector, and z which is orthogonal to v. From our definition of z, we have from a simple rearrangement

u = u_v + z

Basically we can then square the norms of everything on both sides from the Pythagorean theorem which is pic related. Basically we technically have a sum on both sides. So we take the norm and square it (left hand side of the equation), then from the right hand side (usually we call it RHS) we can just expand it out.

Using various properties of dot products we can rearrange this to get Cauchy-Schwarz. Do let me know if you'd like me to clarify any of this. :)

So we can see that u subscript v is just a vector that is useful for us to get u in a form that is nice to play with. In a more intuitive sense u_v is the magnitude of the dot product between u and v, in the direction of v.

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