0:15 Reminder : Lin Properties & subspaces 1:50 Subspaces for geometric vectors 2:28 Rn 6:249:13 Common template for Lin Properties in Rn 7:42 False Lin Properties 9:33 More than one Lin property at the same time 11:05 IMPORTANT: a combination of two Lin Properties is a Lin propert, i.e. the intersection of two subspaces (which results in another subspace)
It is interesting the professor distinguishes between R^n and geometric vectors (plus the origin). I like this point of view. Math professors and linear algebra books try to force the isomorphism between R^n and geometric vectors, but they should be treated as their own objects with their own peculiar behaviors. Of course R^n and geometric vectors share the property that they obey the 'linear property'. Which is, if you add two things in the space or scale something in the space by a number, the result is still in the space.
Again a very nice video! Thanks and please keep up the good work! I feel the professor really likes the subject and teaching it, too. I don't know the name of the prof but is it possible I hear a very faint german accent? Maths rules!!
I dug around and found out that he has a PhD from MIT (well done nice job good game). I olso picked up on the boston accent, but his name is Pavel, and he does have a european accent, so I'm guesing not US born.
Thank you for these excellent videos :-). Why did you mentioned linear combination of coefficients equal to zero rather than linear combination of entries equal to zero? After all you are using entries a, b, and c ! Is there something especial about saying coefficient rather entries?
Monjur Morshed a, b and c are abstractions of the sets of coefficients. Each set of quantities (each vector) is an abstraction of the coefficients of several expressions with corresponding row entries together representing one expression. The entire system of matrix representation in this context is actually a system of abstraction which simplifies notation, but therefore contains much more information than what we are accustomed to in the more popular forms of mathematical expression.
Question: The first type (b=0) makes it so that b is independent of a and c. The second type (b=5a) makes it so that b is dependent on a/a is dependent on b and c is independent of either. The third type (b=(a+c)/2) makes all values (a,b,c) dependent on each other. Does this translate into any specific properties? I know that it affects a geometric interpretation of their subspaces, but I'm still new to this so I don't know of many ways in which these properties can affect them in general.
I think you need a grasp of span and dimension to answer your question. Recall that in R3 space, the only valid subspaces are the zero-vector, R1, and R2. When there are 2 independent vectors (like in these examples), the space they will span is R2 (a plane passing through the origin). So any vector that lies in that plane is reachable by some combination of two vectors on the right hand side. The specific algebraic properties mentioned in the video are additional constraints on the entries of the left-hand side that must be satisfied or there will be no solution (ie no combination of the right hand side will make the equations true).
Any linear combination of ax+by+cz=0 will be a plane in 3D. So if all the vectors share one of linear equation, they all will be on a plane, and if they share two of linear equations, they will be on a line.
11:05 And since both those properties are equal to 0, they can in fact be combined into one ;) 3a - c = 0 = b Then one can rearrange it and get: 3a - b - c = 0, which is, again, a linear property :) And the name "linear" might not be coincidental: these are all *linear equations* (homogeneous ones), that is, they describe a line, plane or flat space going through the origin when we vary the a,b,c (the entries on the list), which confirms that whatever the list describes, must lay on the same line, plane or in the same flat space as every other list fulfilling the same linear property ;) There's a little problem with your definition, though: We can always make the following linear property: 0·a + 0·b + 0·c = 0, and it will ALWAYS be fulfilled, by ANY combination of a,b,c whatsoever :q I guess the correct one should mention that at least one of those coefficients must be non-zero.
Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.
Beautiful lecture! Feels like being at the best university in the world! Thank you. Wow
he is a hidden gem in a treasure box
I am so glad that I found this playlist
0:15 Reminder : Lin Properties & subspaces
1:50 Subspaces for geometric vectors
2:28 Rn
6:24 9:13 Common template for Lin Properties in Rn
7:42 False Lin Properties
9:33 More than one Lin property at the same time
11:05 IMPORTANT: a combination of two Lin Properties is a Lin propert, i.e. the intersection of two subspaces (which results in another subspace)
Awesome 🙂
It is interesting the professor distinguishes between R^n and geometric vectors (plus the origin). I like this point of view. Math professors and linear algebra books try to force the isomorphism between R^n and geometric vectors, but they should be treated as their own objects with their own peculiar behaviors. Of course R^n and geometric vectors share the property that they obey the 'linear property'. Which is, if you add two things in the space or scale something in the space by a number, the result is still in the space.
isomorphism is not the word that describes the relationship between Rn and geometric vectors
Genious!
Again a very nice video! Thanks and please keep up the good work! I feel the professor really likes the subject and teaching it, too. I don't know the name of the prof but is it possible I hear a very faint german accent? Maths rules!!
It's north eastern united states. faintly Boston imho
I dug around and found out that he has a PhD from MIT (well done nice job good game). I olso picked up on the boston accent, but his name is Pavel, and he does have a european accent, so I'm guesing not US born.
do you have to say 'linear' subspace, isn't it enough to say subspace of a vector space, since we are not interested in non linear subspaces.
I agree, "subspace' is sufficient!
Thank you for these excellent videos :-). Why did you mentioned linear combination of coefficients equal to zero rather than linear combination of entries equal to zero? After all you are using entries a, b, and c ! Is there something especial about saying coefficient rather entries?
Monjur Morshed a, b and c are abstractions of the sets of coefficients. Each set of quantities (each vector) is an abstraction of the coefficients of several expressions with corresponding row entries together representing one expression. The entire system of matrix representation in this context is actually a system of abstraction which simplifies notation, but therefore contains much more information than what we are accustomed to in the more popular forms of mathematical expression.
Question: The first type (b=0) makes it so that b is independent of a and c. The second type (b=5a) makes it so that b is dependent on a/a is dependent on b and c is independent of either. The third type (b=(a+c)/2) makes all values (a,b,c) dependent on each other. Does this translate into any specific properties?
I know that it affects a geometric interpretation of their subspaces, but I'm still new to this so I don't know of many ways in which these properties can affect them in general.
+Phoenix Fire I'm not sure. Sometimes it's nice to think algebraically.
I think you need a grasp of span and dimension to answer your question. Recall that in R3 space, the only valid subspaces are the zero-vector, R1, and R2. When there are 2 independent vectors (like in these examples), the space they will span is R2 (a plane passing through the origin). So any vector that lies in that plane is reachable by some combination of two vectors on the right hand side. The specific algebraic properties mentioned in the video are additional constraints on the entries of the left-hand side that must be satisfied or there will be no solution (ie no combination of the right hand side will make the equations true).
Any linear combination of ax+by+cz=0 will be a plane in 3D. So if all the vectors share one of linear equation, they all will be on a plane, and if they share two of linear equations, they will be on a line.
11:05 And since both those properties are equal to 0, they can in fact be combined into one ;) 3a - c = 0 = b
Then one can rearrange it and get: 3a - b - c = 0, which is, again, a linear property :)
And the name "linear" might not be coincidental: these are all *linear equations* (homogeneous ones), that is, they describe a line, plane or flat space going through the origin when we vary the a,b,c (the entries on the list), which confirms that whatever the list describes, must lay on the same line, plane or in the same flat space as every other list fulfilling the same linear property ;)
There's a little problem with your definition, though: We can always make the following linear property: 0·a + 0·b + 0·c = 0, and it will ALWAYS be fulfilled, by ANY combination of a,b,c whatsoever :q I guess the correct one should mention that at least one of those coefficients must be non-zero.
3a - c = 0 = b is a subset of 3a - b - c = 0.
I think you mean "linear combination of entries" by saying "linear combination of coefficients"
Why do we care about subspaces. What is their significance?
Great question! Hoping to record a video about it soon!