Linear Algebra-Subspace Functions

[micromass]

Indeed, so we have the function $f(x)=0$ for all $x$. Does this function satisfy $f(x)=f(-x)$?

 

[FinalStand]

yes. f(x)+0(x) = f(-x) + 0(x)?

 

[Mark44]

yes. f(x)+0(x) = f(-x) + 0(x)?

You don't seem very certain. micromass's question was this:

If f(x) = 0 for all real x, is f(-x) = f(x) for all real x? There is no addition involved.

 

[Fredrik]

yes. f(x)+0(x) = f(-x) + 0(x)?

When f is defined by f(x)=0 for all x in ℝ, as in micromass's post, then yes, we have f(x)=f(-x) for all x. So the equality in the quote above holds. But I'm not sure how the conversation drifted into this.

You seem to understand that there's an f in F(ℝ) such that f+g=g+f=g for all g in F(ℝ). You also seem to understand that an f in F(ℝ) such that f+g=g+f=g for all g in F(ℝ) is the zero vector of F(ℝ). You seem to understand that the f in F(ℝ) defined by f(x)=0 for all x in ℝ has this property. But you still don't seem to have realized that this specific f is the zero vector.

In other words, the f defined by f(x)=0 for all x in ℝ, is the zero vector of F(ℝ), and can therefore be denoted by 0.

Now all you have to do to find out if this specific f is in W, is to determine if f(x)=f(-x) for all x in ℝ.

 

[FinalStand]

I thought f(x) was the vector ? So do I have to say f(x) = 0 is the zero vector and belongs to the subspace since f(x)=f(-x)? I am not exactly sure how to answer this question. How is f a vector? I am so confused, this is so different from what we learned before I took this course. It is more abstract than I like it to be.

 

[Fredrik]

...what is the point on giong over the definitions? And what's the point of proofing this? I think it is all rubbish.

Linear algebra is along with the basics of calculus the most important topic in mathematics. It's used extensively in all sorts of applications of mathematics from quantum mechanics to computer graphics. It's also used in other areas of mathematics. It's a foundation on which many other things are built.

I would define Linear algebra as the subset of the mathematics of linear maps between vector spaces that can be dealt with without using point-set topology (limits of sequences, continuity of functions, etc.) (An equally reasonable but slightly different definition is that it's the mathematics of linear maps between finite-dimensional vector spaces). If you don't know how to check if a given set with an addition operation and a scalar multiplication operation is a vector space, then you will find everything else in linear algebra and its applications very hard or impossible to understand.

So this is as far from being pointless/rubbish/useless/worthless as anything in mathematics can get.

 

[FinalStand]

i was frustrated sorry about that

 

[FinalStand]

So my last response what do you have to say in the answer? DO you just have to say the zero vector is in f(x)=f(-x) as f(x)=0? I just don't know how to shwo your work? As I am not good with showing every step, like you said you have to say "for all x" or w.e. I lost all my marks on communications in Calculus as well because I am nto good at stating variables and working with them.

 

[Fredrik]

I thought f(x) was the vector ? So do I have to say f(x) = 0 is the zero vector and belongs to the subspace since f(x)=f(-x)? I am not exactly sure how to answer this question. How is f a vector? I am so confused, this is so different from what we learned before I took this course. It is more abstract than I like it to be.

f(x) is not a vector, it's a number in the range of the function f, which is a vector. f is called a vector because it's a member of a vector space.

You really need to include the words "for all" where it's appropriate. Note e.g. that when I wrote f(x)=f(-x) at the end of post #54, I made it clear that I was talking about a specific f (the zero vector) and all x. If I had only written "f(x)=f(-x)", that wouldn't have been clear.

The exact statement "f(x)=0 is the zero vector" doesn't really make sense, but I know what you mean. This is a good way to say it: "The function $f:\mathbb R\to\mathbb R$ defined by f(x)=0 for all x in ℝ, is the zero vector of F(ℝ)". It's also OK to just say that the zero vector of F(ℝ) is the function that takes every real number to 0.

To prove that this function is the zero vector, you must pick a notation for it, for example f, and then show that this f satisfies f+g=g for all g in F(ℝ). This is how you do that: Let g be an arbitrary member of F(ℝ). Since f and g both have the domain ℝ, so does f+g. For all x in ℝ, we have (f+g)(x)=f(x)+g(x)=0+g(x)=g(x). This implies that f+g=g.

To prove that this f is in W, all you have to do is to prove that for all x in ℝ, we have f(x)=f(-x). This is how you do that: Let f be the zero vector of F(ℝ). For all x in ℝ, we have f(x)=0=f(-x).

Since it's standard to denote the zero vector of any vector space by 0, I think it's also fine to simplify that last proof to this: For all x in ℝ, we have 0(x)=0=0(-x). This does however look a bit weird, since the first and the last 0 denote the zero vector while the one in the middle denotes the number 0.

Yes, this is pretty abstract. But things will get a lot more abstract as you study more math. This is only the beginning. You just have to get used to it.

 

[FinalStand]

Ok, I was confused on what you guys meant by "vector" I thought the entire f(x) is considered a vector. and when it comes to the zero vector I was confused on whether the "f(x)" is the vector or the (x), so I was completely off. If this continues I think I am going to kill my brain, maybe I need to go take the lectures to make these easier.

 

[Fredrik]

You probably just need to pay more attention to the definitions, be more careful with your statements, and do more exercises. Getting a copy of the textbook would probably be a good start.

 

[FinalStand]

I have always been lazy to the "statements", never thought they were really important, unless it is the conclusion other than that the answer is pretty obvious. In calculus questions for integrals and derivative questions like rate of change and etc. I never put Let x be that so I almost always lost marks on that. Because with all the assigning variables sometimes confuses me on what they mean and confused them with the important variables. I am still thinking in a concrete way so I just go by formula. I really need to get used to this because I am taking a course that has a lot of Proofing next. and my english isn't as good either :P