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# Linearity and Vectors

Modified 2018-06-22 by Andrea Censi

Jacopo Tani

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Linear algebra provides the set of mathematical tools to (a) study linear relationships and (b) describe linear spaces. It is a field of mathematics with important ramifications.

Linearity is an important concept because it is powerful in describing the input-output behavior of many natural phenomena (or systems). As a matter of fact, all those systems that cannot be modeled as linear, still can be approximated as linear to gain an intuition, and sometimes much more, of what is going on.

So, in a way or the other, linear algebra is a starting point for investigating the world around us, and Duckietown is no exception.

This chapter is not intended to be a comprehensive compendium of linear algebra.

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Real numbers are complex for you?: Number theory addref

$\forall$ is a typo for A and $\in$ are Euros? Mathematical symbolic language.

## Linearity

Modified 2018-06-22 by Andrea Censi

In this section we discuss vectors, matrices and linear spaces along with their properties.

Before introducing the these arguments, we need to formally define what we mean by linearity. The word linear comes from the latin linearis, which means pertaining to or resembling a line. You should recall that a line can be represented by an equation like $y = mx + q$, but here we intend linearity as a property of maps, so there is a little more to linearity than lines (although lines are linear maps indeed).

To avoid confusions, let us translate the concept of linearity in mathematical language.

Linearity A function $f: \aset{X} \to \aset{Y}$ is linear when, $\forall x_i \in \aset{X}$, $i = \{1,2\}$, and $\forall a \in \reals$:

\begin{align} f(ax_1) &= af(x_1), \label{eq:lin1}\tag{2} \quad \text{and:} \\ f(x_1 + x_2) &= f(x_1) + f(x_2) \label{eq:lin2}\tag{3} \end{align}

Condition \eqref{eq:lin1} is referred to as the property of homogeneity (of order 1), while condition \eqref{eq:lin2} is referred to as additivity.

Superposition Principle Conditions \eqref{eq:lin1} and \eqref{eq:lin2} can be merged to express the same meaning through: \begin{align} f(ax_1 + bx_2) = af(x_1) + bf(x_2), \forall x_i \in \aset{X}, i = \{1,2\}, \forall a,b \in \reals \label{eq:linearity}\tag{4}. \end{align}

This equivalent condition \eqref{eq:linearity} is instead referred to as superposition principle, which unveils the bottom line of the concept of linearity: adding up (equivalently, scaling up) inputs results in an added up (equivalently, scaled up) output.

## Linear dependance

Modified 2018-06-22 by Andrea Censi

Linear dependance Two or more vectors $\{\avec{v_1},\cdots,\avec{v_n}\}$ are linearly dependant if there exists a set of scalars $\{a_1, \cdots, a_k\}, k \leq n$, that are not all zero, such that: $$\label{eq:lin-dep}\tag{1} a_1\avec{v_1} + \cdots + a_k\avec{v_k} = \avec{0}.$$

When \eqref{eq:lin-dep} is true, it is possible to write at least one vector as a linear combination of the others.

Linear independance Two or more vectors ${\avec{v_1},\cdots,\avec{v_n}}$ are linearly independant if \eqref{eq:lin-dep} can be satisfied only by $k=n$ and $a_i =0, \forall i = 1, \cdots, n$.

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