Literally the best Lin. Algebra video I've seen. I appreciate the pauses in your speech as well. Helped me clarify things in my head without having to rewatch. Thank you so much!
Thanks! Different goals - math textbooks need to be correct and complete. That's not how people tend to learn - the best way is to talk to an expert, who can direct you to what you need and get to the point/intuition.
Thanks! Lifting and jiu-jitsu really don't mix, so I haven't tried putting up huge numbers in years. When I did, 425/350/610, all natural. I never could get deadlifts going due to height and a ligament tear in my upper back. I'm more into functional strength training; my best is 350 burpees in 45 minutes.
Your video is clear and concise, not to mention helpful. It's people like you who make the internet a wonderful tool for learning. Keep up the good work.
Thank you, sir! Since we got to vector spaces, I have understood almost nothing of what my professor says in class, and I so appreciate people like you who do this. I definitely would not be doing nearly as well as I am in this class without people like you.
I like how technical these videos are because my profs ONLY explain in abstract terms taking it hours and days to come up with such techniques with only the given information.
I"m taking a Linear Analysis class and needed a very clear example of the image/range/column space of a matrix and couldn't find one online that didn't confuse me. This was a very VERY straightforward and clear example. Exactly what I was looking for. Thank you!
even though u scared the shit out of me while you were lecturing this subject, i have totally understand the point that you were telling. thank you for your video
My grades have been saved since I discovered your videos last summer retaking Calc III, and once again saved me in Linear Algebra. I can litterally grasp a months worth of classes in 12 hours with your videos and improve my grades dramatically. Thank you so much for putting these videos up. Clearest explanations I've ever seen with any form of math since high school.
Thank you for the video professor. Linear algebra started out so clean and simple. Once spanning, domains, range, kernels and the rest of the mumbo jumbo came into play, I went downhill. I think I needed to start with your videos and others from the beginning. Why I think the professor and the book are enough I'll never know. Old habits die hard.
Your welcome! It really depends on the teacher - if I have a class full of engineers, I'm going to go lighter on the proofs and abstract nonsense. LA is often the first place students encounter proofs in a meaningful way and it can be jarring.
MathDoctorBob This is so true. My class is about 90% Computer Scientists and my professor insists on doing mostly proofs. This example has helped me a ton. Thank you!
+Caden Barton I'm in the exact same situation. Professor has taught everything by proof's and doesn't understand why the class average is a D. Proofs are not how you introduce a topic.
Im taking linear along with differential equations. I much prefer diff eqs... passing one with flying colors the other im holding on for dear life. Same professor for both.
thank you very much for helping me understand what i need to do in my homework! I understand this way better than just reading through my whole textbook. Mahalo!
Same thing, just looked at my review sheet for a final and just did the WTF!?!, spent a few hours watching these videos and now I know what im suppose to be doing vs reading a textbook and having absolutely no clue what its talking about. Thank you!
Thanks! I needed this. Still not certain if I will pass my exam, but you already helped me loads by explaining this! Will definitely check out more of your videos!
@MrKh0i Edit: My bad! I've been out of the weight room for too long. Our workouts were built around bench press, squats, and deadlifts with a lot of isolation work; more geared towards bodybuilding than powerlifting. We just wanted to get big, not necessarily strong or pretty. And nothing formal; we learned everything from Weider and Schwartzenegger's books back in the 80s. - Bob
@FaiththeHairstylist Yes. Kernel of linear transformation T = null space of matrix A for the transformation. A basis is enough to describe the subspace; the entire subspace is given by taking all linear combinations of the basis vectors. - Bob
We should use column operations since row operations do not preserve the column span. The pivots in REF can be used to test for linear dependence. The columns of A are linearly dependent if we can find a nonzero v with Av=0. Going to REF is the same as multiplying by an invertible matrix R on the left, so RAv = 0. If we throw away the non-pivot columns, then RA'v=0 only occurs when v=0. Removing R, what remains is linearly independent.
Good stuff, taking linear over the summer (bad idea), I keep searching my problems, and your videos keep showing up. Good thorough explanations I can google apart if I don't understand a step. Thank you, and subbed. Hopefully you'll be a help when I do Cal 3 in the fall.
You're welcome! It will depend on the domain vector space in general. Suppose we are only using polynomials. We have a linear transformation to R. If we apply the transform to each x^n, we get 1/(n+2). By linearity, each (n+2)x^n - (n+1)x^{n-1) is in the kernel for n=1, 2, . If we add f=1, we have a basis for polynomials, so, without f=1, we have have a basis for the kernel. I can explain more if needed.
Once you find row echelon form, identify the columns with the pivots (pivot - first 1 in a row if any). Use the same columns in the original matrix to get a basis for the range space. Row operations mess up the column span, but the pivots can still be used.
Your welcome! Yes. You can use any nonzero real number. To interpret, you are taking the solution with x3=1 and rescaling the vector by your choice of scalar.
Thanks! Check in the Linear Algebra playlist (or the website). Another way to say isomorphism is invertible or one-one onto. Some videos that will help with this are Example of Basis for Null Space, Linear Trans: One-one, Linear Trans: Onto, and anything on inverting matrices.
awesome video, it really saved me from all these work we gotta do by hand, my teacher mentioned row reduced but he did it another method -_-''...this is a much easier way. Thanks life saver!!
Thanks Dr. Bob! The word "Kernel" was not sitting well with me which was not allowing me to grasp this concept until now-- Ker(T) = Null(A). Simple as that!
You said x1 and x2 are dependant, but if they are written as identity matrix, arent they linearly independent? Plus, the rank of the columns/rows is 2, therefore there should be 2 linearly independent eqautions... Great teaching btw, very straightforward!
I think when he says that x3 is independent, he means that x3 is a free variable. And that the coefficients for x1 and x2 depend on what you choose x3 to be. I don't believe that he means that x1 and x2 are linearly dependent, but that the coefficients to get the null space are dependent on what you choose for x3.
Look into the wrestler's bridge. It's a bodyweight exercise, but just 10 seconds will feel like at eternity. If that doesn't kill you, the gymnast's bridge is the next level of difficulty, but no neck work in it. With weights, shrugs and the neck machine. If you just want stability or to work out aches from sleeping on it wrong, you can do isometric pushing while watching TV.
So excited to get a bigger neck. I feel like my neck is disproportionate to my head and shoulders and this is exactly what I'm looking for. @Dr. Bob you clearly know what you're talking about so I look forward to trying this out!
Thanks! I didn't know there were other methods, but it sounds like the difference is in the bookkeeping. Is this for the kernel or range or both? - Bob
That's it. I would call the pivot columns (1 0 0 0) and (1 1 0 0) because they actually have the pivots in them. A basis for the column span is v1 = (1 2 5 7), v2 = (2 5 14 18). Note that (1 3 9 11) = v2 - v1.
@Pasgo523 You're welcome, and thanks for the comment. Hope the exam went well. If you need linear problems not in the playlist, let me know. I'm a SUNY alum (Stony Brook), so I'm glad to be able to return the favor. - Bob
@israeldmx2003 You're welcome, and thanks for the comment! Yes. It is confusing, but sensible. After all, a linear transformation is not a matrix, but they are as close as possible in spirit. Column space and null space are specific terms for matrices; kernel and range are general algebraic terms that reappear in Abstract Algebra. Good luck on exams! - Bob
They can never be disjoint since 0 is in every subspace. Orthogonal is not even true - consider projections like P(x,y,z)=(x-y,x-y,0). In this case, both the kernel and range are spanned by (1,1,0).
Thank you for being so clear! I hate how my math book is riddled with all these theorems and proofs and barely offers any simple explanations, graphical representations or examples. How hard is it to say that the kernel is the span of the null space, or that the range is the span of the column space? Thanks again.
@kev121314 I just caught this. I will put it in the queue. If this video makes sense, it is just translating. 1-1: null space is zero, or pivot in each column onto: range spans the image R^n , or pivot in each row -Bob
@jonp1101 Thanks for the comment. I used to work for a few SUNY-B alums, and they were really cool. Please let me know if you have any linear algebra requests, and good luck on finals!
My LA playlist is mostly old exam problems with solutions, so it doesn't lend itself to numbeing. Check the website. Everything is listed in order there.
Thanks for the constructive feedback! Every day I learn new tricks, and that's a common request. Of course it doesn't help with the old videos; I could annotate formulas in. Although slightly inconvenient, the pause button works too.
Great job! Very clear, and very organized! Thank you so much! Keep it up Dr. Bob! One Question though.... When it comes to the independent variable x3, can I always use 1? Can I use any number?
@epocalipticify You're welcome! Different modes. Books are better for reference and thoroughness. With a video, I want to get across intuition and why you should care. - Bob
![Noob To Max With DRAGON REWORK In Blox Fruits [FULL MOVIE]](http://i.ytimg.com/vi/LnBMOinoOvA/mqdefault.jpg)
this guy's like "i'll teach the shit out of you!"
Among other things.
Dr.bob looks like a ufc fighter lol
I feel like he looks similar to Channing Tatum
Jiu-jitsu. Actually worked with Pete Spratt in Dallas for a class or two.
Then there's always the Magic Mike route if teaching doesn't work out. :)
LOL. Something tells me your teaching career is going to work out perfectly fine. Thanks for the vids, keep up the excellent work!
Literally the best Lin. Algebra video I've seen. I appreciate the pauses in your speech as well. Helped me clarify things in my head without having to rewatch. Thank you so much!
Your videos make linear algebra seem much easier than many of my textbooks would suggest. What a treasure I have found in these
Thanks! Different goals - math textbooks need to be correct and complete. That's not how people tend to learn - the best way is to talk to an expert, who can direct you to what you need and get to the point/intuition.
Thanks! Lifting and jiu-jitsu really don't mix, so I haven't tried putting up huge numbers in years. When I did, 425/350/610, all natural. I never could get deadlifts going due to height and a ligament tear in my upper back.
I'm more into functional strength training; my best is 350 burpees in 45 minutes.
Thanks for the kind words! I'm just glad to be of help.
"God made the integers; all else is the work of man." - Kronecker.
Your video is clear and concise, not to mention helpful. It's people like you who make the internet a wonderful tool for learning. Keep up the good work.
Can't believe your methods are still so helpful and effective even after ten years! Thank you, sir! Wish u have a great time teaching!
Thank you, sir! Since we got to vector spaces, I have understood almost nothing of what my professor says in class, and I so appreciate people like you who do this. I definitely would not be doing nearly as well as I am in this class without people like you.
I like how technical these videos are because my profs ONLY explain in abstract terms taking it hours and days to come up with such techniques with only the given information.
I"m taking a Linear Analysis class and needed a very clear example of the image/range/column space of a matrix and couldn't find one online that didn't confuse me. This was a very VERY straightforward and clear example. Exactly what I was looking for. Thank you!
Your welcome and good luck on finals!
I like your method. Basically the working is left entirely to us, and you go though it all to guide us. Stay blessed.
even though u scared the shit out of me while you were lecturing this subject, i have totally understand the point that you were telling. thank you for your video
My grades have been saved since I discovered your videos last summer retaking Calc III, and once again saved me in Linear Algebra. I can litterally grasp a months worth of classes in 12 hours with your videos and improve my grades dramatically. Thank you so much for putting these videos up. Clearest explanations I've ever seen with any form of math since high school.
Thank you for the video professor. Linear algebra started out so clean and simple. Once spanning, domains, range, kernels and the rest of the mumbo jumbo came into play, I went downhill. I think I needed to start with your videos and others from the beginning. Why I think the professor and the book are enough I'll never know. Old habits die hard.
Your welcome! It really depends on the teacher - if I have a class full of engineers, I'm going to go lighter on the proofs and abstract nonsense. LA is often the first place students encounter proofs in a meaningful way and it can be jarring.
MathDoctorBob This is so true. My class is about 90% Computer Scientists and my professor insists on doing mostly proofs. This example has helped me a ton. Thank you!
+Caden Barton I'm in the exact same situation. Professor has taught everything by proof's and doesn't understand why the class average is a D. Proofs are not how you introduce a topic.
+Scot Matson yeah our prof shows us like 2 worked easy problems before the proofs.
helps so much
thank you ffs finally someone that keeps the explanation simple, you looking pretty buff bro tbh
You're welcome! Yeah, this stuff is hard enough with clear communication. - Bob
If you don't understand this, you will get hit by the math stick of justice!
Lmao
Perhaps he was in the military at one point
Math stick of truth. Justice has a moral dimension.
I loved math until I took Linear Algebra.
tried multivariable calculus? it's killing me, but is also quite fun though
Im taking linear along with differential equations. I much prefer diff eqs... passing one with flying colors the other im holding on for dear life. Same professor for both.
Yep same! I’m taking that with calc and I’m questioning why I’m a math major 😂
One of the better videos I've come across, had to leave you a comment.
thank you very much for helping me understand what i need to do in my homework! I understand this way better than just reading through my whole textbook. Mahalo!
Perfect video, the only one I could find on youtube that directly addresses how to find the range and null space. I think I'm going to watch it again!
Incredibly well explained - thank you
Your welcome! Good luck with finals.
You're welcome, and thanks for the high praise! Glad to be of help. - Bob
Same thing, just looked at my review sheet for a final and just did the WTF!?!, spent a few hours watching these videos and now I know what im suppose to be doing vs reading a textbook and having absolutely no clue what its talking about. Thank you!
Feels like my dad is explaining me this, like a ticking bomb about to explode any minute.
Rorschach test!
cleared range and kernel a lot right before exam....thankss
Your welcome! Good luck on finals
THANK YOU! I'll get at least one problem right on my test tomorrow...
:D
Thank you Sir.. You helped me by filling gaps in my understanding
Really old video, yet it's the most efficient and easy
Thanks! - Really old Bob :)
idk why i laught hella hard when he raised his hand and was holding a damn pole
lol
finished my last problem with your help. thanks Dr. Bob
Thanks! I needed this. Still not certain if I will pass my exam, but you already helped me loads by explaining this! Will definitely check out more of your videos!
Your welcome! Good luck on exams!
This actually makes sense. Nice work
I understood from those 7 minutes what my professor couldn't explain in 2 hours.
THANK YOU
You're welcome, and thanks for the kind words! - Bob
@MrKh0i Edit: My bad! I've been out of the weight room for too long. Our workouts were built around bench press, squats, and deadlifts with a lot of isolation work; more geared towards bodybuilding than powerlifting. We just wanted to get big, not necessarily strong or pretty. And nothing formal; we learned everything from Weider and Schwartzenegger's books back in the 80s. - Bob
@FaiththeHairstylist Yes. Kernel of linear transformation T = null space of matrix A for the transformation. A basis is enough to describe the subspace; the entire subspace is given by taking all linear combinations of the basis vectors. - Bob
We should use column operations since row operations do not preserve the column span.
The pivots in REF can be used to test for linear dependence. The columns of A are linearly dependent if we can find a nonzero v with Av=0. Going to REF is the same as multiplying by an invertible matrix R on the left, so RAv = 0. If we throw away the non-pivot columns, then RA'v=0 only occurs when v=0. Removing R, what remains is linearly independent.
Good stuff, taking linear over the summer (bad idea), I keep searching my problems, and your videos keep showing up. Good thorough explanations I can google apart if I don't understand a step. Thank you, and subbed. Hopefully you'll be a help when I do Cal 3 in the fall.
You're welcome! It will depend on the domain vector space in general.
Suppose we are only using polynomials. We have a linear transformation to R. If we apply the transform to each x^n, we get 1/(n+2). By linearity, each (n+2)x^n - (n+1)x^{n-1) is in the kernel for n=1, 2, . If we add f=1, we have a basis for polynomials, so, without f=1, we have have a basis for the kernel. I can explain more if needed.
Once you find row echelon form, identify the columns with the pivots (pivot - first 1 in a row if any). Use the same columns in the original matrix to get a basis for the range space. Row operations mess up the column span, but the pivots can still be used.
this was definitely very helpful! thanks for uploading this video!
Your welcome! Yes. You can use any nonzero real number. To interpret, you are taking the solution with x3=1 and rescaling the vector by your choice of scalar.
my guy look like he playin Clue, "it was Dr Bob in the Foyer with the lead pipe"
Colonel Mustard brought it on himself.
@@MathDoctorBob haha! for real though, great video, it was a huge help!
Great!
Also like that wooden stick you are holding to! Look way more cool than an ordinary "pointer".
And now I got away with bunking lectures and thanks to you Dr Bob, will surely get it right on my exam #Brilliant
I get the impression he's holding back serious rage whilst delivering his points. Great video Bob anyhow.
tylerdurden786 yeah, he is gonna beat you up with his black stick if you fail your exam
@@h-grid3137 BAHAHAHA
Thank you, Dr.Bob! Very very helpful review video before finals! Keep it up!
Your welcome! Good luck on finals!
Thank you so much! I have a understood the entire concept clearly!
Thanks! Check in the Linear Algebra playlist (or the website). Another way to say isomorphism is invertible or one-one onto. Some videos that will help with this are Example of Basis for Null Space, Linear Trans: One-one, Linear Trans: Onto, and anything on inverting matrices.
great video, thank you! I understand them so well
awesome video, it really saved me from all these work we gotta do by hand, my teacher mentioned row reduced but he did it another method -_-''...this is a much easier way. Thanks life saver!!
That was awesome.thanks.please do more videos for linear.
Most intimidating teacher of all time. Muscles and a bar.
@finapon You're welcome, and thanks for the support! Good luck on exams. If you have any specific requests, please let me know. - Bob
Buffest math teacher I've ever seen.
Thanks Dr. Bob! The word "Kernel" was not sitting well with me which was not allowing me to grasp this concept until now-- Ker(T) = Null(A). Simple as that!
Erick Q Technically we use kernel for linear transformations and null for matrices, but even this is not ironclad.
You're welcome, and thanks for the kind words! Without the pauses, I tend to speak in paragraphs.
thxs very much Dr Bob....I finally got it...really appreciate it
Thanks for the kind words!
cool stuff,thanks Bob for the video. I found it helpful
You said x1 and x2 are dependant, but if they are written as identity matrix, arent they linearly independent? Plus, the rank of the columns/rows is 2, therefore there should be 2 linearly independent eqautions... Great teaching btw, very straightforward!
Yeah that was a mistake x1 and x2 are independent. x3 is dependent.
I think when he says that x3 is independent, he means that x3 is a free variable. And that the coefficients for x1 and x2 depend on what you choose x3 to be. I don't believe that he means that x1 and x2 are linearly dependent, but that the coefficients to get the null space are dependent on what you choose for x3.
@MetallicAus You're welcome. Thanks for the comment. - Bob
Great work teacher
dr bob, any neck workout tips?
Look into the wrestler's bridge. It's a bodyweight exercise, but just 10 seconds will feel like at eternity. If that doesn't kill you, the gymnast's bridge is the next level of difficulty, but no neck work in it. With weights, shrugs and the neck machine. If you just want stability or to work out aches from sleeping on it wrong, you can do isometric pushing while watching TV.
So excited to get a bigger neck. I feel like my neck is disproportionate to my head and shoulders and this is exactly what I'm looking for. @Dr. Bob you clearly know what you're talking about so I look forward to trying this out!
Thanks! I didn't know there were other methods, but it sounds like the difference is in the bookkeeping. Is this for the kernel or range or both? - Bob
Thanks a lot.....I was confused in that particular part.
Your video really helped, thanks!
That's it. I would call the pivot columns (1 0 0 0) and (1 1 0 0) because they actually have the pivots in them. A basis for the column span is v1 = (1 2 5 7), v2 = (2 5 14 18). Note that (1 3 9 11) = v2 - v1.
@Pasgo523 You're welcome, and thanks for the comment. Hope the exam went well. If you need linear problems not in the playlist, let me know. I'm a SUNY alum (Stony Brook), so I'm glad to be able to return the favor. - Bob
super helpful vid, good shit!
@israeldmx2003 You're welcome, and thanks for the comment! Yes. It is confusing, but sensible. After all, a linear transformation is not a matrix, but they are as close as possible in spirit. Column space and null space are specific terms for matrices; kernel and range are general algebraic terms that reappear in Abstract Algebra.
Good luck on exams! - Bob
Great explanation. Thank you.
very good and well organized. thank you.
Your videos are awesome,
but seriously what I think everyone wants to know is: how much do you bench, deadlift and squat?
They can never be disjoint since 0 is in every subspace. Orthogonal is not even true - consider projections like P(x,y,z)=(x-y,x-y,0). In this case, both the kernel and range are spanned by (1,1,0).
this guy is like the drill instructor of maths teachers
Thank you for being so clear! I hate how my math book is riddled with all these theorems and proofs and barely offers any simple explanations, graphical representations or examples. How hard is it to say that the kernel is the span of the null space, or that the range is the span of the column space? Thanks again.
Very clearly explained, thanks very much
Great video! It helped me a lot :D
@kev121314 I just caught this. I will put it in the queue. If this video makes sense, it is just translating.
1-1: null space is zero, or pivot in each column
onto: range spans the image R^n , or pivot in each row
-Bob
@jonp1101 Thanks for the comment. I used to work for a few SUNY-B alums, and they were really cool. Please let me know if you have any linear algebra requests, and good luck on finals!
this is perfect, thank you :)
My LA playlist is mostly old exam problems with solutions, so it doesn't lend itself to numbeing. Check the website. Everything is listed in order there.
really really nice explanation. Thank u sir
NICE ONE SIR. VERY INTUITIVE
THANK YOU! very helpful!
great explanation, thanks a ton!
Thanks for the constructive feedback! Every day I learn new tricks, and that's a common request. Of course it doesn't help with the old videos; I could annotate formulas in. Although slightly inconvenient, the pause button works too.
PERFECTION MAN PER-FEC-TION:) I LOVED THIS:) KEEP GOING:)
you are amazing!!!! after listening to it three times i understand everything. do you have any video on geometric transformations from R2 TO R2?
Thanks! Yes, it's called Linear Transformations on R2. You can find the full set of links for Linear Algebra at mathdoctorbob dot org.
coming in clutch! thank you
if he was my prof I may shit myself each time I go to his lecture... but still, you helped me a lot thank you very much
Great job! Very clear, and very organized! Thank you so much! Keep it up Dr. Bob!
One Question though.... When it comes to the independent variable x3, can I always use 1? Can I use any number?
Good video! Thank you so much!
@epocalipticify You're welcome! Different modes. Books are better for reference and thoroughness. With a video, I want to get across intuition and why you should care. - Bob
Dr bob. first of all ur biceps are huge man. second, your one effin kickass prof
:) Good luck on finals!
@lordcroesus You're welcome! Please let me know if you have any requests. - Bob
better than my lecturer hahah thanks!!!!!!!! :)
Good and understandable :) great job