If you know what happens to basic pieces, then you know what happens to any combination of those pieces, because the combinations are formed only by scaling and adding. It means complex behavior can be reconstructed from simpler components without distortion of the combination rules. This is why linear systems are tractable. They preserve structure.

Linear example

$$f(x) = 3x$$

Then:

$$f(x + y) = 3(x + y) = 3x + 3y = f(x) + f(y)$$

and

$$f(\alpha x) = 3\alpha x = \alpha \cdot 3x = \alpha f(x).$$

So this is linear.

Nonlinear example

$$f(x) = x^2$$

Then:

$$f(x + y) = (x + y)^2 = x^2 + 2xy + y^2$$

which is not usually $f(x) + f(y)$.

The extra $2xy$ term is the hallmark of nonlinearity here. The combination produces interaction terms not already present in the original pieces.

That is a good intuitive marker: nonlinearity introduces new interactions between parts.

Links to this note:
Vector Space

By a vector space we mean a nonempty set $E$ with two operations:

This just means there is some collection of objects, called $E$, and it is not empty. Why “nonempty”? Because a vector space must at least have something in it. In fact, the rules will force it to contain a special element called the zero vector. The elements of $E$ are called vectors. Important: at this stage, “vector” does not mean “arrow in space.” It just means “element of this set $E$.” It could be: arrows, tuples like $(𝑥,𝑦,𝑧) (x,y,z)$, functions, polynomials, matrices, sequences, and so on. So “vector” here means “thing that behaves according to these rules.”

\((x, y) \mapsto x + y\) from \(E \times E\) into \(E\), called addition,

This says: Take any two vectors $𝑥$ and $𝑦$ from $𝐸$, and there is a rule that produces another element $𝑥+𝑦$, also in $𝐸$. What does “from $𝐸×𝐸$ into $𝐸$” mean? $E×E$ means all ordered pairs $(𝑥,𝑦)$ with both $𝑥$, $𝑦$ $∈$ $𝐸$. “into $𝐸$” means the result is again an element of $𝐸$.
So addition is a function:input: two vectors,output: one vector.
This is the closure idea: adding vectors keeps you inside the space

$(\lambda, x) \mapsto \lambda x$ from $\mathbb{F} \times E$ into $E$, called multiplication by scalars,

This says: Take a scalar $𝜆$ from $𝐹$, and a vector $𝑥$ from $𝐸$, and the rule gives another vector $𝜆𝑥$ in $𝐸$. Here $𝐹$ is the field of scalars, $F=R$ for real numbers, or $𝐹=𝐶$ for complex numbers.
So scalar multiplication means: numbers can “act on” vectors.
Again, the result must stay in $𝐸$.

Big picture so far

A vector space is: a set $𝐸$ of vectors, an addition rule for vectors,a scalar multiplication rule using numbers from $𝐹$, with some laws (coming up). Those laws are what make the structure behave like ordinary linear geometry.

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• What two operations are necessary for vector space?
  Addition and Scalar Multiplication.
• Why are these two operations included in vector space: addition and scalar multiplication?
  Addition and scalar multiplication are singled out because together they generate all linear combinations.
• What happens to vector space if you add more operations?
  You get different structures.
• In vector space if you add {{c1::a notion of length}}, you get a {{c2::normed vector space}}.
• In vector space if you add {{c1::angles and orthogonality}}, you get an {{c2::inner product space}}.
• In vector space if the {{c1::inner product space is complete}}, you get a {{c2::Hilbert space}}.

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Such that the following conditions are satisfied for all $x, y, z \in E$ and $\alpha, \beta \in \mathbb{F}$:

$x + y = y + x$;

This is commutativity of addition. It says the order of addition does not matter. You want vector addition to behave symmetrically. No vector should get privileged treatment just because it was written first.

$(x + y) + z = x + (y + z)$;

This is associativity of addition.It says when adding vectors, grouping does not matter. This means you can write $𝑥+𝑦+𝑧$ without ambiguity.

For every $x, y \in E$ there exists a $z \in E$ such that $x + z = y$;

Most textbooks split this into: existence of a zero vector, existence of additive inverses $x+(-x)=0$. But this axiom packages both ideas into one condition. Given any starting vector $𝑥$ and any target vector $𝑦$, there is some vector $𝑧$ that takes you from $𝑥$ to $𝑦$ by addition. So you can always “solve for the difference” between two vectors. If you set $𝑦=𝑥$, then this condition says there exists some $𝑧$ such that $𝑥+𝑧=𝑥$. This gives something behaving like a zero vector relative to $𝑥$. And from the other axioms, one can prove there is a unique common zero vector for all vectors. Then, setting $y=0$, this condition gives a vector $𝑧$ such that $𝑥+𝑧=0$. That $𝑧$ is the additive inverse of $𝑥$, usually written $−𝑥$.

$\alpha(\beta x) = (\alpha \beta)x$;

This is compatibility of scalar multiplication with field multiplication. It says multiplying by $𝛽$, then by $𝛼$, is the same as multiplying once by the product $𝛼𝛽$.

$(\alpha + \beta)x = \alpha x + \beta x$;

This is one distributive law. It says if you add two scalars first and then multiply the vector, that is the same as multiplying separately and then adding. This makes scalar multiplication linear in the scalar.

$\alpha(x + y) = \alpha x + \alpha y$;

This is the other distributive law. It says scalar multiplication distributes over vector addition. This makes scalar multiplication linear in the vector.

$1x = x$.

This says multiplying by the scalar 1 does nothing. So the multiplicative identity of the field acts trivially on vectors. Without this, scalar multiplication would not properly match the meaning of ordinary multiplication.

What is this definition really trying to capture?

It is trying to isolate the essence of linearity.

A vector space is any setting where:

• you can add “states,” “directions,” or “quantities,”
• you can scale them,
• and these operations behave coherently.

That is why the same definition applies to arrows in geometry, solutions to differential equations, signals, functions, matrices, and quantum states.

The power comes from abstraction:
once something satisfies these rules, all linear methods become available.

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• Addition and scalar mutliplication is necessary for vector spaces but alone insufficient. That is why we needs the axioms that follow. Can you explain why?
  Addition and scalar multiplication tell you what kinds of moves are allowed.
  The axioms tell you how to constrain the behaviour so it counts as genuinely linear.
• How many axioms are there?
  7

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The Math Only

By a vector space we mean a nonempty set $E$ with two operations:

• $(x, y) \mapsto x + y$ from $E \times E$ into $E$, called addition,
• $(\lambda, x) \mapsto \lambda x$ from $\mathbb{F} \times E$ into $E$, called multiplication by scalars,

such that the following conditions are satisfied for all $x, y, z \in E$ and $\alpha, \beta \in \mathbb{F}$:

1. $x + y = y + x$;
2. $(x + y) + z = x + (y + z)$;
3. For every $x, y \in E$ there exists a $z \in E$ such that $x + z = y$;
4. $\alpha(\beta x) = (\alpha \beta)x$;
5. $(\alpha + \beta)x = \alpha x + \beta x$;
6. $\alpha(x + y) = \alpha x + \alpha y$;
7. $1x = x$.

Elements of $E$ are called vectors. If $\mathbb{F} = \mathbb{R}$, then $E$ is called a real vector space, and if $\mathbb{F} = \mathbb{C}$, $E$ is called a complex vector space. (Source: Introduction to Hilbert spaces with applications, Lokenath Debnath)

Summary

You could read the definition as:
$E$: the set of vectors.
$F$: the scalars.
$x+y$: vector addition.
$\lambda x$: scalar multiplication.

Rules:
(a) order of addition does not matter,
(b) grouping of addition does not matter,
(c) subtraction is always possible,
(d) repeated scaling matches scalar multiplication,
(e) scaling is distributive over scalar addition,
(f) scaling is distributive over vector addition,
(g) multiplying by 1 does nothing.