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I'm currently stuck in two vector space problems. I feel as though the signs and my lack of knowledge throw me off. I am not aware of what $\boxplus, \boxdot$ mean in the first problem (should I just think of them as addition and multiplication). Same happens with problem 2. Should I assume $\oplus, \odot $ mean addition and multiplication. Please help, thank you.

Problem 1: In ${R}^2$, consider the following operations:

$(x_1, y_1) \boxplus (x_2, y_2) = $(max {$x_1, x_2$}, max { $y_1, y_2$ })

$\alpha \boxdot (x,y) = (\alpha * x, 0)$

is ${R}^2$ with these operations a vector space? If your answer is negative, list all the vector spaces axioms that fail to be satisfied and explain why; otherwise prove that all the axioms are satisfied.

Problem 2: In ${R}^2$, consider the following operations:

$(x_1, y_1) \oplus (x_2, y_2) = (x_1 + x_2, 0) \alpha \odot (x,y) = (\alpha * x, y) $

is ${R}^2$ with these operations a vector space? If your answer is negative, list all the vector spaces axioms that fail to be satisfied and explain why; otherwise prove that all the axioms are satisfied.

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  • $\begingroup$ Yes, they means addition and scalar multiplication (but not the usual ones, and this notation is introduced to be less confusional than the ordinary + and ·). $\endgroup$ – Bernard Sep 7 '19 at 22:12
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    $\begingroup$ "I am not aware of what $\boxplus,\boxdot$ mean in the first problem..." It literally defines what $\boxplus,\boxdot$ mean right there in the problem statement... That is what the lines are doing where the symbols first appear. They define $\boxplus$ to be the binary operation over $\Bbb R^2$ such that if you combine two elements of $\Bbb R^2$ via $\boxplus$ the result is as written. Namely, $(x_1,y_1)\boxplus (x_2,y_2) = (\max\{x_1,x_2\},\max\{y_1,y_2\})$ $\endgroup$ – JMoravitz Sep 7 '19 at 22:17
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Recall that a vector space $V$ over $F$ is a set together with an operation that takes two elements of $V$ and gives you an element of $V$, which we call the “sum” of the two elements; and an operation that takes an element of $F$ and an element of $V$ and gives you an element of $V$, which we call the “scalar product”.

These operations need not be related to what we usually call “sum” and “product”. In order to avoid possible confusion with operations we usually call sum and product, we may want to use different symbols.

For example, we usually define “the sum of $(a,b)$ and $(c,d)$” to be the vector $(a+c,b+d)$, where the sum is the usual sum of real numbers. But we don’t have to define it this way; we could try to come up with a different way of defining it. So in order to prevent us from confusing this new way of “adding” pairs with the usual way, we use a different symbol, so as to keep it separate. Since $+$ denotes the usual sum of real numbers, instead we will use a symbol which is sufficiently similar to remind us it’s supposed to be a “sum”, but sufficiently different to remind us it is not the usual sum. Common choices are $\boxplus$ and $\oplus$.

So you define the way to “combine” two vectors $(a,b)$ and $(c,d)$ to get a new vector, called “$(a,b)\boxplus (c,d)$”, using the definition $$(a,b)\boxplus(c,d)\text{ will represent the pair }(\max{a,c}, \max{b,d}).$$ This may or may not satisfy the conditions we need to have for this way of “combining” vectors to work as the sum in a vector space... you are being asked to check whether it does.

Similarly, $\boxdot$ is just a definition of a potential “scalar multiplication” that you need to check to see whether it satisfies the requirements to be the scalar multiplication of a vector space.

And again, same thing with $\oplus$ and $\odot$ in problem 2: these are the definitions, and you should check if these operations “work” to get a vector space.

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  • $\begingroup$ Thank you for this explanation it was very concise. I'll try to work the problems now. $\endgroup$ – Josue Sep 7 '19 at 22:21
  • $\begingroup$ @Josue: If you find an answer helpful, remember to upvote it (you may be unable to do so until you get enough reputation); if your query has been answered, remember to accept the answer you find most satisfying, so that the question can be marked as “answered” in the site. $\endgroup$ – Arturo Magidin Sep 7 '19 at 22:23
  • $\begingroup$ I will as soon as I hit the reputation points. $\endgroup$ – Josue Sep 7 '19 at 22:28
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The sign $(x_1,y_1)\boxplus (x_2,y_2)$ between the two pairs of numbers defines an new kind of addition same with the sign $\boxdot$ in $\alpha\boxdot(𝑥,𝑦)=(𝛼𝑥,0)$a new kind of multiplication by a scalar is defined, you have to find out if with these new definitions in $\mathbb{R}^2$ you have still a Vector space, if not, what axioms do not hold.

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