I Closure in the subspace of linear combinations of vectors

Click For Summary
The discussion centers on the concept of closure in vector subspaces, particularly when combining vectors from different dimensions. It questions why closure is lost in the third subspace formed by linear combinations of the first two subspaces, which are defined in the x_hat and y_hat directions. The confusion arises from the assumption that adding vectors from these subspaces would still yield a vector that is a linear combination of the original vectors. However, the clarification suggests that the phrasing of the question may have led to misunderstanding, indicating that the issue might be more about language than mathematical principles. Ultimately, the conversation highlights the importance of precise language in discussing mathematical concepts.
Phys12
Messages
351
Reaction score
42
TL;DR
Definition 12 in Principles of Quantum Mechanics, it says that when you have a subspace of vectors in 1 dimension, and another subspace of vectors in another dimension and finally a 3rd subspace with the linear combination of the first two; in the last case, closure will be lost. Why is that?
1595278278010.png


This is the exact definition and I've summarized it, as I understand it above. Why is it, that for elements in the third subspace, closure will be lost? Wouldn't you still get another vector (when you add two vectors in that subspace), that's still a linear combination of the vectors in the first two subspaces? So, if the first subspace is of all vectors in the x_hat direction and the second subspace is of vectors in the y_hat direction. Then the third subspace pretty contains all the vectors in 2D and when you add two of them, it will give you another, that can always be written as a linear combination of x_hat and y_hat. What am I missing here?
 
Physics news on Phys.org
Phys12 said:
Summary:: Definition 12 in Principles of Quantum Mechanics, it says that when you have a subspace of vectors in 1 dimension, and another subspace of vectors in another dimension and finally a 3rd subspace with the linear combination of the first two; in the last case, closure will be lost. Why is that?

View attachment 266632

This is the exact definition and I've summarized it, as I understand it above. Why is it, that for elements in the third subspace, closure will be lost? Wouldn't you still get another vector (when you add two vectors in that subspace), that's still a linear combination of the vectors in the first two subspaces? So, if the first subspace is of all vectors in the x_hat direction and the second subspace is of vectors in the y_hat direction. Then the third subspace pretty contains all the vectors in 2D and when you add two of them, it will give you another, that can always be written as a linear combination of x_hat and y_hat. What am I missing here?

I think this is a question about English, not math. "But for the elements (3)" should be read as "If not for the elements (3)".
 
  • Like
Likes fresh_42
George Jones said:
I think this is a question about English, not math. "But for the elements (3)" should be read as "If not for the elements (3)".
Ah, I see, thanks! LOL
 

Thread 'Angular Momentum Vector and Its Magnitude'

For the quantum state ##|l,m\rangle= |2,0\rangle## the z-component of angular momentum is zero and ##|L^2|=6 \hbar^2##. According to uncertainty it is impossible to determine the values of ##L_x, L_y, L_z## simultaneously. However, we know that ##L_x## and ## L_y##, like ##L_z##, get the values ##(-2,-1,0,1,2) \hbar##. In other words, for the state ##|2,0\rangle## we have ##\vec{L}=(L_x, L_y,0)## with ##L_x## and ## L_y## one of the values ##(-2,-1,0,1,2) \hbar##. But none of these...
View full post »

Similar threads

I Vector subspace and basis vectors in the context of data science
  • · Replies 6 ·
Replies
6
Views
3K
I Smallest subspace if a plane and a line are passing through the origin
  • · Replies 2 ·
Replies
2
Views
2K
Help with linear algebra: vectorspace and subspace
  • · Replies 15 ·
Replies
15
Views
3K
I Quantum Superposition, Linear Combinations and Basis
  • · Replies 1 ·
Replies
1
Views
1K
I Finding a linear combination to enter a sphere
  • · Replies 3 ·
Replies
3
Views
1K
How to show a subspace must be all of a vector space
  • · Replies 8 ·
Replies
8
Views
2K
Dimension of orthogonal subspaces sum
  • · Replies 4 ·
Replies
4
Views
2K
I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
  • · Replies 3 ·
Replies
3
Views
3K
MHB Are These Sets Vector Subspaces of R^3?
  • · Replies 5 ·
Replies
5
Views
2K
I Affine hull and affine combinations equivalence
  • · Replies 7 ·
Replies
7
Views
3K