# Exploring Linear Space in the Real & Complex Planes

• Numeralysis
In summary, the conversation discusses the definition of a Linear Space and the properties that define it, such as v \in V then 1v = v or \vec{0} \in V such that \vec{0} + v = v. These properties may break down in the complex plane, but they are included in the definition because they are useful and necessary for many theorems to hold true. The conversation also mentions that every object introduced as a linear space has these properties and it is rare to find a space that satisfies all vector space axioms except for these two.
Numeralysis
I was wondering, some of the things that define a Linear Space such as:

$$v \in V$$ then $$1v = v$$ or $$\vec{0} \in V$$ such that $$\vec{0} + v = v$$

They seem very obvious and intuitive, but, is there ever a time they break down in the Real plane? I think they might break down in the complex plane, but, I'm not too sure how they would.

What do you mean, break down?

The two pretty obvious & intuitive properties. Where they don't work anymore; such that when you have v living in V and you multiply 1 by v, it longer equals v or add the zero vector it doesn't equal itself?

Then you don't have a vector space.

Exactly, but, I'm wondering when does this definition not hold true. These two properties seem pretty obvious, and pretty intuitive. More than anything, I'm wondering why are they included when defining a Linear Space. Other than for extra-proofing.

And if these definitions fail.

They are included because they are useful; there are many interesting theorems about vector spaces, and there are lots of things that can be modeled by vector spaces. If you omit some axioms such as 1v = v or 0 + v = v, then many of the theorems fail to be true.

They seem pretty intuitive, because every object that has been introduced to you as a linear space has those properties. If you have a set with an operation, we almost always use 0 and 1 to be defined as the additive and multiplicative identities; so if you had a set with an operation that had no such identity, we wouldn't call elements 0 and 1.

Off the top of my head I'm not able to think of a space that satisfies every vector space axiom except for those two.

The statements 0+v= v and 1v=v are really saying that there are additive and multiplicative identities and telling you particularly what they are (0 & 1).

## 1. What is linear space?

Linear space, also known as vector space, is a mathematical concept that refers to a set of objects (called vectors) that can be added together and multiplied by scalars (such as numbers) to create new vectors. In simpler terms, it is a collection of objects that can be manipulated using basic mathematical operations.

## 2. What is the difference between real and complex planes?

The real plane, also known as the Cartesian plane, is a two-dimensional coordinate system that uses two real numbers (x and y) to locate a point on a graph. The complex plane, on the other hand, uses two complex numbers (a + bi) to locate a point, where a and b are real numbers and i is the imaginary unit. Essentially, the complex plane is an extension of the real plane.

## 3. How are vectors represented in the real and complex planes?

In the real plane, vectors are represented as arrows with a specific length and direction. In the complex plane, vectors are represented as points (also known as complex numbers) with a specific magnitude and angle. The magnitude of the vector is represented by the distance from the origin and the angle is represented by the direction from the positive real axis.

## 4. What is the significance of exploring linear space in the real and complex planes?

Exploring linear space in the real and complex planes allows us to understand and analyze the properties and behaviors of vectors in a visual and geometric way. This can help us solve complex mathematical problems and applications in various fields such as physics, engineering, and computer science.

## 5. How can one apply the concepts of linear space in the real and complex planes?

The concepts of linear space in the real and complex planes can be applied in various ways, such as solving systems of linear equations, modeling physical systems, and transforming data in computer graphics. It is also used in advanced topics in mathematics, such as linear algebra and complex analysis.

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