Vector Subspaces: Determining U as a Subspace of M4x4 Matrices

[mathiebug7]

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

 

[fresh_42]

What are upper symmetric matrices? Doesn't sound symmetric though. And what do you think? You should show us your work!

 

[FactChecker]

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.

 

[vela]

What are upper symmetric matrices? Doesn't sound symmetric though. And what do you think?

I'm guessing diagonal matrices.

 

[fresh_42]

I'm guessing diagonal matrices.

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!

 

[Orodruin]

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 …

 

[FactChecker]

I'm guessing diagonal matrices.

That certainly would work. This gets my vote.

 

[WWGD]

Ultimately, OP needs to show operational closure under addition, scaling.

 

[FactChecker]

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.