Determine is the set is a real vector space (and say why if it isn't)

[Pi Face]

-set of all nonnegative real numbers
-set of all upper triangular nxn matrices
-set of all upper triangular square matrices

from the book, the answers are: yes, no(need to be the same size to define addition), and yes

for the set of all nonnegative real numbers, doesn't it fail closure under scalar multiplication? if you multiply any number (except 0) in the set by a negative number, making it a negative number, wouldn't that new number be out of the initial set? thus making it not a vector space?

for upper triangular nxn matrices, why isn't it a vector space? nxn implies they are all the same size right? so when you add two of them together, you get another upper triangular nxn matrix, and multiplying them preserves this as well.

for all upper triagnular square matrices, why is it a real vector space? the set just specifices "square", not the size, so if you have two different sized triaangular matrices then you can't add them, failing closure under addition

 

[Pi Face]

anyone have any idea?

 

[sunjin09]

I think you are right and the book is wrong, but since it's in a book, I could be missing something