Vector Subspaces: Determining U as a Subspace of M4x4 Matrices

In summary, the document explores the concept of vector subspaces within the context of \( M_{4x4} \) matrices. It focuses on the criteria for determining whether a given subset \( U \) of matrices qualifies as a subspace. Key properties such as closure under addition and scalar multiplication are examined, along with examples that illustrate the conditions needed for \( U \) to be considered a subspace of the larger matrix space. The analysis highlights the importance of these properties in linear algebra and their applications in various mathematical contexts.
  • #1
mathiebug7
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Determine whether the following subsets U of M4x4is a subspace of the vector space V of all M4x4 matrices, with
the standard operations of matrix addition and scalar multiplication. If is not a subspace provide an example to
demonstrate a property that U does not possess.
a. The set U of all 4x4 upper symmetric matrices
 
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  • #2
What are upper symmetric matrices? Doesn't sound symmetric though. And what do you think? You should show us your work!
 
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  • #3
mathiebug7 said:
a. The set U of all 4x4 upper symmetric matrices
I am familiar with "upper triangular" and "symmetric", but I don't know what "upper symmetric" is.

In any case, you should proceed step-by-step through the definition of a vector space and show whether or not each requirement is satisfied. If any of them are difficult, we can give hints and guidance on textbook-type problems where you show your work.
 
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  • #4
fresh_42 said:
What are upper symmetric matrices? Doesn't sound symmetric though. And what do you think?
I'm guessing diagonal matrices. :wink:
 
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  • #5
vela said:
I'm guessing diagonal matrices. :wink:
This could be a way out if there wasn't that "U" and the extensive description of how to find a counterexample. Very suspicious, Watson!
 
  • #6
fresh_42 said:
This could be a way out if there wasn't that "U" and the extensive description of how to find a counterexample. Very suspicious, Watson!
The OP seems to have posed only the (a) part of the question. Presumably the b, c, etc parts have been left out …
 
  • #7
vela said:
I'm guessing diagonal matrices. :wink:
That certainly would work. This gets my vote.
 
  • #8
Ultimately, OP needs to show operational closure under addition, scaling.
 
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  • #9
WWGD said:
Ultimately, OP needs to show operational closure under addition, scaling.
Good point. IMO, for a beginning student, it would be good for him to go down the properties and list the ones that are inherited. Then prove the closure properties that are not just inherited.
 
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FAQ: Vector Subspaces: Determining U as a Subspace of M4x4 Matrices

What is a vector subspace?

A vector subspace is a subset of a vector space that is also a vector space under the same operations of addition and scalar multiplication. This means it must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication.

How do you determine if U is a subspace of M4x4 matrices?

To determine if U is a subspace of M4x4 matrices, you must verify three conditions: (1) U contains the zero matrix, (2) U is closed under matrix addition, and (3) U is closed under scalar multiplication. If all these conditions are satisfied, then U is a subspace of M4x4 matrices.

What is the zero matrix in M4x4, and why is it important for subspaces?

The zero matrix in M4x4 is the 4x4 matrix where all entries are zero. It is important for subspaces because the presence of the zero matrix is a necessary condition for any subset of M4x4 to be considered a subspace. Without the zero matrix, the subset cannot satisfy the requirements of a vector space.

Can you provide an example of a subspace of M4x4 matrices?

One example of a subspace of M4x4 matrices is the set of all 4x4 diagonal matrices. This set includes the zero matrix, is closed under matrix addition (the sum of two diagonal matrices is also a diagonal matrix), and is closed under scalar multiplication (a scalar times a diagonal matrix is also a diagonal matrix).

What is the significance of closure properties in determining subspaces?

The closure properties ensure that the operations of addition and scalar multiplication do not produce elements outside the subset. This is crucial because for a subset to be a subspace, it must behave consistently with the rules of the larger vector space. Without closure, the subset would not retain the structure necessary to be considered a vector space in its own right.

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