• tobor8man
In summary, the book "Reading in Mathematical Physics" by Sadri Hassani defines a vector as an abstract element with certain properties such as addition and multiplication by scalars. This definition also extends to other mathematical objects such as scalars, matrices, and tensors. In the context of this definition, a scalar is a vector and a matrix is a vector, but a tensor is a vector of a different rank. It is important to note that these objects are defined by their transformation rules and can belong to different vector spaces.
tobor8man

There are these things called vectors, a, b, x etc., that have these properties:

a + b = b + a
a + (b + c) = (b + a) + c
a + 0 = a, 0 is the zero vector
a + (- a) = 0

You can multiply them by complex numbers (scalars) like c, d
c(d a) = (cd)a
1 a = a

Multiplication involving vectors and scalars is distributive
c(a + b) = c a+ c b
(c + d) a = c a+ d a

And that is it.

Given that definition, a scalar is a vector, a matrix is a vector, a tensor is a vector. Yes?

Mind you, I have also read that scalars and vectors are a kinds of tensors, of rank 0 and 1 respectively. True?

Am I confused? Should I be?

tobor8man said:

There are these things called vectors, a, b, x etc., that have these properties:

a + b = b + a
a + (b + c) = (b + a) + c
a + 0 = a, 0 is the zero vector
a + (- a) = 0

You can multiply them by complex numbers (scalars) like c, d
c(d a) = (cd)a
1 a = a

Multiplication involving vectors and scalars is distributive
c(a + b) = c a+ c b
(c + d) a = c a+ d a

And that is it.

Given that definition, a scalar is a vector, a matrix is a vector, a tensor is a vector. Yes?

Mind you, I have also read that scalars and vectors are a kinds of tensors, of rank 0 and 1 respectively. True?

Am I confused? Should I be?

you shouldn't be confused... you'll learn that first) ther's not only euclidean geometry----> different metrics----> then a vector is a generalization of a tensor, you'l learn the meaning of covarianca and controvariance... and you'll understand that everything id defined by its tranformation rule...

say an euclidean vector V is something that |V|^2 is an invariant under SO(3) and that under rotation transform as: $$V'^{\mu}=R_{\mu\nu}V^{\nu}$$ where R belongs to SO(3).
keep going...
regards
marco

A vector is an element of a vector space. Thus, elements of the real line are vectors if you endow the real line with the algebraic structure of a vector space. The set of all continuous functions over R is also a vector space, where the vectors are functions. Tensors with the usual addition are also elements of a vector space when regarded in that manner. This is a space where the vectors are tensors, and so forth.

## 1. What is a vector?

A vector is a mathematical object that represents both magnitude and direction. It can be thought of as an arrow in space, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

## 2. How is a vector different from a scalar?

A scalar is a mathematical object that represents only magnitude, while a vector represents both magnitude and direction. For example, temperature is a scalar quantity, while velocity is a vector quantity.

## 3. What are some real-life examples of vectors?

Real-life examples of vectors include displacement, velocity, force, and acceleration. For instance, if a car is traveling at 50 miles per hour in a northward direction, the velocity of the car is a vector with a magnitude of 50 miles per hour and a direction of north.

## 4. How are vectors used in physics?

Vectors are used in physics to represent physical quantities that have both magnitude and direction. They are especially useful in describing motion and forces, as they allow us to easily calculate and analyze the various components of these quantities.

## 5. Is it important to understand vectors in mathematics and physics?

Yes, understanding vectors is crucial in mathematics and physics. They are used in many branches of mathematics, such as linear algebra and calculus, and are essential in solving problems in physics, particularly in mechanics and electromagnetism.

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