Dear Dr. Hower, thank you for your motivating (always smiling) lectures. As I am refreshing my Linear Algebra. Thank you again for going an extra mile.
I am self studying linear algebra through the internet. As a person who is studying to know linear algebra it's self not to get a certain degree or some special document ,this have been helpful to me honestly it boosted my studying spirit ,Thank you so much❤❤
Good day Dr.Valerie, Just finished another of your now world famous lectures on Linear Algebra if I also may believe the other positive comments. I would like to add something to your clear presentation if you do not object. You can also distinguish the three possible solutions of a system of linear equations as follows: 1) A system with one unique solution is called consistent and independent...2) A system with infinitely many solutions is called consistent and dependent...3) A system without solution(s) is called inconsistent. Of course I'm fully convinced that you already know this, I just thought maybe naming the terms 'dependent and independent' could be a small but not unimportant addition. So nice to see that such a system can be interpreted in several ways and of which I like the way of a linear combination of the column vectors of a given matrix; fascinating! This was new to me, and it feels more intuitive than the dot product way. By the way Dr. Valerie, I can see that you need to have quite a bit of dancing skills to be able to explain the dot product of vectors; to be honest I'm not a good dancer myself! Dr. Valerie, happy Valentine's Day to the most entertaining teacher on the internet!
Hello. I think you are referring to columns of the coefficient matrix being linearly independent or dependent. This is a good point. I discuss linear independence in a future video.
@@DrValerieHower Dr. Valerie, You are correct about the use of the terms dependent/independent. Indeed it is premature to mention them in this lecture, given that linear (in)dependence of a set of vectors at this stage of the playlist has not been covered by you yet! I apologize for this. Dr. Valerie, have a nice day.
Great video. My book uses A_mxn for the dimensions of the coefficient matrix , so i will use that instead of n x m. I wish they standardized these things.
All those antic over some chalk and a chalk board. Yea I’m learning something. I’m sixty five so it will never mean much despite my computer science and electronics degree. Fifteen minutes a day then I’m going to go over the Krebs and Calvin cycles of biology chemistry wise. That’s why this is so much fun.
Dr. Hower, one thing I am curious about is why texts even mention row echelon form? Why not only talk about *reduced* REF? Maybe I missed this somewhere in your lecture? It seems to me that one would always want to go fully to RREF (assuming the system is consistent of course). What is the value of only REF? Fantastic series. Thank you!
This is a great question that we revisit....the idea of dimension. But, in that example, I had one free variable t and in my solution t could be any real number. This is what makes the line.
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you cant imagine how I am thankful to you. Thank you
You are so welcome. I appreciate the comment :)
Dear Dr. Hower, thank you for your motivating (always smiling) lectures. As I am refreshing my Linear Algebra. Thank you again for going an extra mile.
You are welcome! I really appreciate the feedback :)
We certainly are amused!
I am self studying linear algebra through the internet. As a person who is studying to know linear algebra it's self not to get a certain degree or some special document ,this have been helpful to me honestly it boosted my studying spirit ,Thank you so much❤❤
You are very welcome! I really appreciate the feedback.
I’m really enjoying these lectures. Thank you for all the time and effort you put into these!
You are welcome and thank you for your feedback!
Concepts have been presented in a crystal clear manner. Amazing!
Thank you so much!
Passed my calculus 2 last semester! Thank you for all the help! Now I’m taking linear algebra this summer 😄
You are very welcome! I really appreciate the feedback. I wish you the best. :)
Good day Dr.Valerie, Just finished another of your now world famous lectures on Linear Algebra if I also may believe the other positive comments. I would like to add something to your clear presentation if you do not object. You can also distinguish the three possible solutions of a system of linear equations as follows: 1) A system with one unique solution is called consistent and independent...2) A system with infinitely many solutions is called consistent and dependent...3) A system without solution(s) is called inconsistent. Of course I'm fully convinced that you already know this, I just thought maybe naming the terms 'dependent and independent' could be a small but not unimportant addition. So nice to see that such a system can be interpreted in several ways and of which I like the way of a linear combination of the column vectors of a given matrix; fascinating! This was new to me, and it feels more intuitive than the dot product way. By the way Dr. Valerie, I can see that you need to have quite a bit of dancing skills to be able to explain the dot product of vectors; to be honest I'm not a good dancer myself! Dr. Valerie, happy Valentine's Day to the most entertaining teacher on the internet!
Hello. I think you are referring to columns of the coefficient matrix being linearly independent or dependent. This is a good point. I discuss linear independence in a future video.
@@DrValerieHower Dr. Valerie, You are correct about the use of the terms dependent/independent. Indeed it is premature to mention them in this lecture, given that linear (in)dependence of a set of vectors at this stage of the playlist has not been covered by you yet! I apologize for this. Dr. Valerie, have a nice day.
@@jan-willemreens9010 No apology needed, of course. It was a very good observation!
@@DrValerieHower Dr. Valerie, I appreciate this! Have a good evening
1:00:38 I like how you displayed 3 rref moves and lined them up next to the rows they will replace. Very neat.
thank you for your feedback!
Great video. My book uses A_mxn for the dimensions of the coefficient matrix , so i will use that instead of n x m. I wish they standardized these things.
yes. This depends on the text. Some use m x n and some use n x m. What is important is that you understand the notation is (rows) x (columns).
All those antic over some chalk and a chalk board. Yea I’m learning something. I’m sixty five so it will never mean much despite my computer science and electronics degree. Fifteen minutes a day then I’m going to go over the Krebs and Calvin cycles of biology chemistry wise. That’s why this is so much fun.
I appreciate the comment :). Thanks for sharing!!
Dr. Hower, one thing I am curious about is why texts even mention row echelon form? Why not only talk about *reduced* REF? Maybe I missed this somewhere in your lecture? It seems to me that one would always want to go fully to RREF (assuming the system is consistent of course). What is the value of only REF? Fantastic series. Thank you!
REF takes fewer steps computationally. This may be one reason.
Hello there are tactics for algebra
In previous video at 42:05 you said the vector will be a line. How did you know whether it is a point, line, plane or entire Rn
This is a great question that we revisit....the idea of dimension. But, in that example, I had one free variable t and in my solution t could be any real number. This is what makes the line.
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this is not a question I have thought about before. Best wishes to you!