How do I verify the vector space axioms?

[grandpa2390]

Determine whether the following sets, equipped with the given operations, are vector spaces. Verify all vectorspace axioms for those that are vector spaces; identify the vector space axioms that fail for those that are notvector spaces.
6. The set of all ordered pairs of real numbers with the operations

?, ?

+

?!, ?′

=

?, ? + ?′

and? ?,?

=

??,??

for all real numbers ?.

this isn't the only problem, and I have tried watching videos but the problems are different and I don't understand how to verify the 10 axioms.
could somebody help walk me through this. Not looking for the answer, I want to understand how to do it.

 

[stapel]

Determine whether the following sets, equipped with the given operations, are vector spaces. Verify all vectorspace axioms for those that are vector spaces; identify the vector space axioms that fail for those that are notvector spaces.
6. The set of all ordered pairs of real numbers with the operations

?, ?

+

?!, ?′

=

?, ? + ?′

and? ?,?

=

??,??

for all real numbers ?.

this isn't the only problem, and I have tried watching videos but the problems are different and I don't understand how to verify the 10 axioms.
could somebody help walk me through this. Not looking for the answer, I want to understand how to do it.

I'm sorry, but all of these files appear to be located on your own hard drive. As such, they are not available to us.

Kindly please reply with the typed-out text of the execise and its instructions, along with a complete listing of your thoughts and efforts so far. Thank you!

 

[Ishuda]

Determine whether the following sets, equipped with the given operations, are vector spaces. Verify all vectorspace axioms for those that are vector spaces; identify the vector space axioms that fail for those that are notvector spaces.
6. The set of all ordered pairs of real numbers with the operations

?, ?

+

?!, ?′

=

?, ? + ?′

and? ?,?

=

??,??

for all real numbers ?.

this isn't the only problem, and I have tried watching videos but the problems are different and I don't understand how to verify the 10 axioms.
could somebody help walk me through this. Not looking for the answer, I want to understand how to do it.

A lot of the time the 'proof' comes from the fact that the underlying field has the needed property. For example consider the vector space of the ordered pairs (x,y) where x and y belong to the reals. Then vector addition defined by
\(\displaystyle (x_1, y_1) + (x_2, y_2) = (x_1+x_2, y_1+y_2)\)
works because the reals are closed under addition.

 

[grandpa2390]

I'm sorry, but all of these files appear to be located on your own hard drive. As such, they are not available to us.

Kindly please reply with the typed-out text of the execise and its instructions, along with a complete listing of your thoughts and efforts so far. Thank you!

grrr. I couldn't verify because it had to be approved by moderator. but when I pasted it it appeared correct.

Determine whether the following sets, equipped with the given operations, are vector spaces. Verify all vectorspace axioms for those that are vector spaces; identify the vector space axioms that fail for those that are notvector spaces.


6. The set of all ordered pairs of real numbers with the operations ?, ? + ?!, ?′ = ?, ? + ?′ and? ?,? = ??,?? for all real numbers ?.