Recent content by Austin Chang

How does lemma 18.2.2 imply the definition?

What happened to this friend of yours? I am starting to see a correlation with all of your answers and sorry for me not being super specific with the word "grinding". It is similar to honing your skills in mathematics or any work by doing a lot of problems and figuring ways to improve your...

Hi all. As seen by the title, I am having trouble studying. I am currently a college student and especially since my parents are paying for my tuition, I feel the pressure of doing well. I know I am not allocating enough time to doing well in my classroom and much of that time is spent grinding...

What do you mean by at least a vector space of dimension 1? I don't think I've gotten that far to understand it completely. Could you elaborate?

Thanks! That was a good example.

So for your example if it was not unique U2 = U1 and you can write w1 in U2 and w2 in U1? therefore w2+w1 is not unique anymore because the same things came from different subspaces?

Let V be a vector space. If U 1 and U2 are subspaces of V s.t. U1+U2 = V and U1 and U1∩U2 = {0V}, then we say that V is the internal direct sum of U1 and U2. In this case we write V = U1⊕U2. Show that V is internal direct sum of U1 and U2if and only if every vector in V may be written uniquely...

Ok I see What you mean I guess then it will just mean a lot of work. Thanks!

So for example #1 I have to prove that f+g exists in the Vector Space. I think it looks obvious to most people, but how do I necessarily put in words. Do i just say for all f,g that are elements in our set f'(0) = 0 g'(0) = 0. Therefore (f+g)'(0) = f'(0) +g'(0)= 0+0=0( where do we get the tools...

I totally agree with what you have to say on post #13. I believe I am no longer confuse about what a vector space is. I was confused at first because what I saw in the book was very simple ones that you could picture with your mind and see if it worked, but I now know that for example the one we...

How would we prove that the first 1 seems really trivial i mean you pull out from the same domain and the function will always scale somewhere in the range. The 2 one do you want me to prove this with limits? because that would take a while. 3 one kinda seems obvious from what fresh_42 said...

My biggest question right now is wouldn't showing this true right now would require identity function which we have not proven yet? F'(0) = 0 and g'(0) = 0 (f+g)'(0) = f'(0) +g'(0) which will be distributivity? sorry I'm not sure. But my point is how can we prove that addition is closed in this...

DO we have to show this explicitly with the field the set is in?

I am sorry but I am not totally sure what this means and back to my original question does it mean this is a vector space since the function must have a degree ≠1 in order for it to satisfy the parameter ƒ'(0) = 0. Therefore addition exists because f+g will exist within this space because it...

From my understanding Vector spaces are set in which they follow particular rules such as commutativity, addition, associativity, scalar multiplication, having a zero vector, and identity.