Subspace or not?

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In summary, we are discussing whether the subset S, which includes all invertible 2x2 matrices, is closed under addition and scalar multiplication. While it is true that adding two non-singular matrices may result in a non-invertible matrix, this does not hold true for all invertible matrices. Therefore, we cannot conclude that S is not closed under addition. As for scalar multiplication, we have not discussed it and cannot make any conclusions.
  • #1
Maxwhale
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Homework Statement



S is a subset of vector space V.

If V = 2x2 matrix and S ={A | A is invertible}

a) is S closed under addition?
b) is S closed under scalar multiplication?

Homework Equations





The Attempt at a Solution



For non singular 2x2 matrices, S is not closed under addition. but I am not quite sure about invertible 2x2 matrix.

Say, A = [1 0]
[0 1]

So, if we add A + A, it is still invertible, so it is closed under addition. But does my statement lose the generality?
 
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  • #2
Your statement does lose the generality of the argument. Similarly I could argue if A is

[1 0]
[0 0]

and B is

[0 1]
[0 0]

A+B is non-invertible, hence non-singular matrices are closed under addition. But you know that's false. To prove the set of invertible matrices is closed under addition, you need to prove that given ANY A and B (not just a single example) A + B is invertible. Alternatively, you can find a single counterexample to prove that it is not closed under + (since if there exists A,B such that A+B is not invertible, then it is not true that A+B is invertible for all A,B)
 
  • #3
Maxwhale said:

Homework Statement



S is a subset of vector space V.

If V = 2x2 matrix and S ={A | A is invertible}

a) is S closed under addition?
b) is S closed under scalar multiplication?

Homework Equations




The Attempt at a Solution



For non singular 2x2 matrices, S is not closed under addition. but I am not quite sure about invertible 2x2 matrix.

Say, A = [1 0]
[0 1]

So, if we add A + A, it is still invertible, so it is closed under addition. But does my statement lose the generality?

What's the difference between 'non-singular' and 'invertible'? Aren't they the same thing?
 

1. What is subspace and how is it related to science?

Subspace is a mathematical concept that refers to a subset of a larger mathematical space. It is commonly used in linear algebra and physics to describe a space that is within a larger space. In science, subspace can be used to describe a physical space within a larger system, such as a subspace within a galaxy or a subspace within a cell.

2. How do scientists determine if a space is a subspace or not?

There are several criteria that can be used to determine if a space is a subspace or not. One of the most important criteria is closure, which means that the space must be closed under addition and scalar multiplication. Other criteria include the existence of a zero vector, the existence of inverse elements, and the existence of a basis.

3. Can subspace exist in more than three dimensions?

Yes, subspace can exist in any number of dimensions. In fact, in linear algebra, subspaces are often defined in higher dimensions. However, it can be difficult to visualize subspace in higher dimensions, so it is often described using mathematical equations and properties.

4. How is subspace used in scientific research?

Subspace is used in various fields of science, such as physics, engineering, and computer science. In physics, subspace is used to describe the properties of physical systems, such as energy levels in an atom. In engineering, subspace is used in control systems to describe the behavior of a system. In computer science, subspace is used in machine learning algorithms and data analysis.

5. Are there real-life examples of subspace?

Yes, there are many real-life examples of subspace. In physics, the energy levels of an atom can be described as a subspace within the larger space of the atom. In biology, the genetic code of an organism can be seen as a subspace within the larger space of all possible genetic codes. In technology, the internet can be thought of as a subspace within the larger space of all computer networks.

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