# Linear Algebra - Linear Spaces

• Mano Jow
In summary, you are trying to solve a linear algebra problem, but you have some doubts about the axioms. You are trying to prove that increasing functions are a subset of functions, but you are having trouble with the addition and multiplication operations. You have tried a solution, but it doesn't seem to work.
Mano Jow
Hello there,
I'm having some problems with a proof-like exercise on linear algebra. Here's what I'm supposed to do:

## Homework Statement

Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold.

## Homework Equations

All increasing functions.
(there are others, like rational functions, all Taylor polynomials of degree $$\leq$$ n, etc. but I guess my doubts are related to all of them).

## The Attempt at a Solution

Well, after a couple of attempts my teacher told me that once I prove that increasing functions are a subset of the functions linear space, I just need to prove the closure axioms. This can be done just by saying "the increasing fuctions are contained in the set of functions". Ok, then I started to prove the closure under addtion.

Let f be an increasing funcion, if x1 < x2, then f(x1) < f(x2). The same applies to a increasing function g.
So, adding g(x1) to both sides of the f(x) equation and f(x2) to both sides of the g(x) equation, we have:

(1) f(x1) + g(x1) < f(x2) + g(x1)
(2) g(x1) + f(x2) < g(x2) + f(x2)
$$\Rightarrow$$ f(x1) + g(x1) < g(x2) + f(x2)
$$\Rightarrow$$ (f+g)(x1) < (f+g)(x2)

And I guess it's proven that it's closed under addition. Is that right?
Now, I know it's not closed under multiplication because if I multiplicate f by -1 it won't be an increasing function anymore. But can I just say that and it's ok or is there another way to show it's not closed under multiplication?

I have tried this (let a be a real number):
f(x1) < f(x2) $$\Rightarrow$$ af(x1) < af(x2) $$\Rightarrow$$ (af)(x1) < (af)(x2). But I don't think I got anything useful.

I'd be very grateful if anyone could help.
Thanks in advance.

Mano Jow said:
Hello there,
I'm having some problems with a proof-like exercise on linear algebra. Here's what I'm supposed to do:

## Homework Statement

Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold.

## Homework Equations

All increasing functions.
(there are others, like rational functions, all Taylor polynomials of degree $$\leq$$ n, etc. but I guess my doubts are related to all of them).

## The Attempt at a Solution

Well, after a couple of attempts my teacher told me that once I prove that increasing functions are a subset of the functions linear space, I just need to prove the closure axioms. This can be done just by saying "the increasing fuctions are contained in the set of functions". Ok, then I started to prove the closure under addtion.

Let f be an increasing funcion, if x1 < x2, then f(x1) < f(x2). The same applies to a increasing function g.
So, adding g(x1) to both sides of the f(x) equation and f(x2) to both sides of the g(x) equation, we have:

(1) f(x1) + g(x1) < f(x2) + g(x1)
Since g(x1) < g(x2), you can continue the inequality above with < f(x2) + g(x2).

This shows that f(x1) + g(x1) < f(x2) + g(x2), or
(f + g)(x1) < (f + g)(x2), and you're done with that part.
Mano Jow said:
(2) g(x1) + f(x2) < g(x2) + f(x2)
$$\Rightarrow$$ f(x1) + g(x1) < g(x2) + f(x2)
$$\Rightarrow$$ (f+g)(x1) < (f+g)(x2)

And I guess it's proven that it's closed under addition. Is that right?
Now, I know it's not closed under multiplication because if I multiplicate f by -1 it won't be an increasing function anymore. But can I just say that and it's ok or is there another way to show it's not closed under multiplication?
This is enough to show that the set of increasing functions is not closed under scalar multiplication.
Mano Jow said:
I have tried this (let a be a real number):
f(x1) < f(x2) $$\Rightarrow$$ af(x1) < af(x2) $$\Rightarrow$$ (af)(x1) < (af)(x2). But I don't think I got anything useful.

I'd be very grateful if anyone could help.
Thanks in advance.

Thanks a lot for the fast reply!

## 1. What is a linear space?

A linear space, also known as a vector space, is a mathematical structure that consists of a set of objects (vectors) that can be added together and multiplied by scalars (numbers) to produce new vectors. These objects must also satisfy certain axioms, such as closure under addition and scalar multiplication, in order to be considered a linear space.

## 2. What are some examples of linear spaces?

Some common examples of linear spaces include the set of all real numbers, the set of all polynomials of a certain degree, and the set of all 2D or 3D vectors. Other examples can include spaces of matrices, functions, or sequences.

## 3. What is the difference between a subspace and a span?

A subspace is a subset of a larger linear space that still satisfies the axioms of a linear space. This means that a subspace is a smaller linear space within a larger one. On the other hand, the span of a set of vectors is the set of all possible linear combinations of those vectors, which may or may not be a linear space itself. The span can be thought of as the "space" created by a set of vectors, while a subspace is an actual subset of a linear space.

## 4. How is linear independence related to linear spaces?

Linear independence is a concept that is closely related to linear spaces. It refers to a set of vectors that are not linearly dependent on each other, meaning that none of the vectors in the set can be written as a linear combination of the others. In a linear space, a set of linearly independent vectors can form a basis, which is a set of vectors that can be used to represent all other vectors in the space.

## 5. How is linear algebra used in real life?

Linear algebra has many practical applications in fields such as engineering, physics, computer science, and economics. It is used to solve systems of linear equations, analyze data sets, and model real-world systems. Examples of its applications include image and signal processing, machine learning, and optimization problems.

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