The way R^2 is not a subspace of R^3 makes sense to me is that you could have a plane in R^3 that does not passes through the origin ( meaning it does not contain the zero vector) from looking at the sketch @ 11:40
Even the plane (in R^3) passing through the origin would not be a subspace because it has a value for Z coordinate meanwhile in R^2 there is no such thing. R^2 isn't just R^3 with Z set to zero. Z doesn't exist in R^2. This is why the statement is false.
To be exact, what makes you know that the vector is in the span? what do I have to identify to let me know that? Am I to go over span of vectors first?
Sorry, but is there any word definition of subspace? Like what is it and what is it used for? Because I understand how to do the math but when asked to explain what it is, I would have no idea what to say. If anyone knows, plz let me know. >< Thanks!
This topic is getting into advanced mathematics. Layman's explanations are harder to come up with. You need to understand the notion of sets, vectors, vector spaces to draw conclusions about those mathematical objects and topics, and how they work together. I don't think this is for average people to understand, it's for people who are talented in mathematics. If you need to pass a course in college as part of your degree, zoom in on it and learn how to solve the problems mechanically and ignore context and meta.
@@georghieronymus9935 Instead of writing all that you could have just either said it's not useful for laymen and you won't explain it, or just attempt to explain it.
I think yeah!! what I thought was:- from the last example, it is looking obvious that the subspace and the vector have to possess the same dimensions...
From 12:10 onwards, you said R3 has to give that vector to R2 and therefore R2 is a subset of R3 which sounds similar to the time you drew the diagram at 11:39. That I don't understand.
It is a nice tutor even a university professor doesn't explain like this. But I have some doubt on one of the last three questions, which is question number 3. I think the answer should be TRUE but you said FALSE. Why I said TRUE is, a vector in R2 means that the third axis, which is axis Z is 0 and it is omitted. Therefore, R2 is a subspace of R3. If you or anyone who reading this can explain that my argument is wrong or right. Thank you.
you dont explain much about the why, you do but like not enough to understand :( like 4:30 "we know its in the subspace because it is in the span... like how is it in the span why waaaaaa
Because a span of some vectors is just a linear combination of those vectors, meaning any way you add those vectors together after multiplying them by any real number scalar. Basically span(v1,v2...vn) = a1*v1+a2*v2...+an*vn. You can have any vector and any number of them you want. In that particular case, you got v1 = [-1,1,1], v2=[0,-6,2], v3=[1,0,0], and the scalars a,b,c, which can basically be anything and you add the vectors together after multiplying by the scalars. Therefore, you got a span of v1,v2,v3, which is same as span([-1,1,1], [0,-6,2], [1,0,0]). Since his previous proof in this video showed that span of any number of vectors is a subspace of the vector space that they belong to, that particular form of vector is a subspace to the vector space of column space size 3. Correct me if I'm wrong.
I don't find your explanations to be entirely clear. You're just saying things and writing things down but not saying why. Your explanations are too shallow man.You're losing me on the second example.
