I\'ve got the first part of the proof where you assume W is a subspace and show

Question

I've got the first part of the proof where you assume W is a subspace and show the rest it is llike straight definition. I am having issues with the second direction. I don't understand how we can assume that "ax" is an element of W. Just because "a" is a scalar and "x" is a vector doesn't mean that "ax" is in the subset right? So if you could be specific with that part I would be appreciative. I get the rest just not how you can jump from "ax+y" is in the susbet to "ax" is in the subset. Thanks

Explanation / Answer

In the second direction you must assume that ax + y is in W for ALL a in F and ALL x, y in W. So you can set a = 1 and see that 1x + y = x + y is in W whenever x and y are in W, and set y = 0, which shows that ax + 0 = ax is in W whenever a is in F and x is in W.

I gather from your question that you've done everything else yourself, so please write a comment if something isn't clear to you.

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