Linear Algebra 4h: Unions and Intersections of Linear Subspaces

[darrenpeck]

Please consider covering modules. Would benefit so many people. Thank you!

[lukedavis]

this is the best exposition of this topic I've heard. this was really unnecessarily confusing in David C. Lay and Sheldon Axler. thank you for this!

[rickygallonegro]

I think this is difficult to grasp because we imagine numbers to be super expansive and when we read about the intersection the only example we get is the trivial example. This guy really is a master teacher.

[antonellomascarello]

6:03 Geometric Vectors
8:22 The union is not a subspace
10:17 The intersection is a subspace
14:02 IMPORTANT NOTE

[maxpercer]

pavel i am going to watch all your videos. this is it

[agh]

Your explanation made perfect sense without me having to pause the video to "explain it to myself".

[MarkLeavenworth]

This actually took me nearly an hour to explain to myself, but this thought is what helped me: On union: 'Is it possible for that which admits of only one linear subspace to be in the same linear subspace with that which admits of only being in another linear subspace?...and on intersection: 'Is it possible for that which admits of having two linear subspace properties, or belonging in two linear sub spaces simultaneously to not exist in any linear subspace?'

[xoppa]

13:20 I am following your argument that the intersection of two subspaces is also a vector space, the vector sum part.
How is this argument.
Let A, B be vector subspaces of vector space V.
Show that v1 + v2 ∈ A ∩ B whenever v1, v2 ∈ A ∩ B.

Proof:
1. v1, v2 ∈ A ∩ B implies v1, v2 ∈ A and v1, v2 ∈B.
2. v1, v2 ∈A implies v1 + v2 ∈A (since we assumed A is a subspace).
3. v1, v2 ∈B , then v1+ v2 ∈B (since we assumed B is a subspace).
4. By steps 2 & 3, v1 + v2 ∈A and v1 + v2 ∈B.
5. By the definition of set intersection, v1 + v2 ∈A ∩ B .

[xoppa]

Just a clarification, the way you defined vectors the zero vector is always at the origin (or a dot at the origin). That way vectors are always emanating from the origin. Or to put it another way, if you shrink vectors they will vanish at the origin.

[boutiquemaths]

Exciting end to the section! The playlist is missing a section 5? Will we be skipping any content if we start watching the videos in section 6?

[ImprEcr]

Ok ... I'm on point . Thanks by the way, beautiful.

[kunal]

In the geometric vector space, consider a plane parallel to x-y plane as a subspace, and y-z plane as another subspace, their intersection will be a line. My question is if we select any vector from that line, multiplied by 0, we will get zero vector, which does not belong to that subspace. So, does that mean intersection of two subspaces is not linear subspace?

[alekssandroassisbarbosa]

is possible to see the exercises before the video solution?