Motivation for definition of linearity

Question

I recently started studying linear algebra, and I learnt that the definition of linear function is different than that of elementary algebra. T is linear if:

• T(a + b) = T(a) + T(b)
• T(cv) = c T(v) is what I learnt. I’ve been trying to find a motivation for defining it like this for weeks, but I can’t find one. I really want to understand this as I believe it’s extremely important in understanding ‘linear’ algebra. (I have only been exposed to vector spaces, linear independence, bases, and elementary algebra) (I also do not see why we’d like to ‘preserve the structure of vector spaces’.)

Answer 1

Studying linear algebra you surely have noticed the importance of the basis for a vector space. Intuitively a basis describe an entire vector space just with a smaller collection of this vectors. Since every vector $v$ in a vector space $V$ over a field $\mathbb{K}$ given a base of V say $\{v_1,v_2...,v_n\}$ could be written as $$v = a_1v_1+a_2v_2+ ... +a_nv_n$$ with $a_i \in \mathbb{K}, \space\forall i = 1, 2,...n$.

Linearity is a huge help because knowing how a certain linear application say $L$ acts on the base of the vector space, tells you how $L$ acts on every single other vector, in fact by definition $$L(v) = L ( a_1v_1+a_2v_2+ ... +a_nv_n) = a_1L(v_1)+a_2L(v_2)+ ... +a_nL(v_n)$$ This is just an example of why linearity is defined as it is, clearly there are some other reason much more deeper not even I fully understood. Hope this can help.

Answer 2

There's not one answer to your question, but I'll try my best to try to make clear the importance of certain aspects.

So first of all, vector spaces are a type of structure that we really like and they pop-up in a lot of applications. Vector spaces are fairly simply, a vector space is a set $V$ and the elements of this set are called vectors. A vector space has two meaningful operation, a plus (+) and a scalar multiplication. By this we mean that you add two vectors $v$ and $w$ to get a new vector $v+w$ and given any real number $\lambda$, you can multiply a vector by this number to get a new vector $\lambda\cdot v$.

With these two operations, the expression $\lambda v+\mu w$ makes sense whenever $v,w\in V$ and $\lambda,\mu\in \mathbb{R}$. Such an expression is called a linear combination (of two vectors, you can guess what a linear combination of more vectors is). So to summarize this part, a vector space is a structure in which linear combinations make sense.

As an example, $\mathbb{R}^2=\left\{\begin{pmatrix}x\\y\end{pmatrix}\mid x,y\in \mathbb{R}\right\}$ is the set of coordinates of the plane. Given two coordinates, is clear how to add them and multiplication by a number is also clear, namely $$\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}z\\t\end{pmatrix}=\begin{pmatrix}x+z\\y+t\end{pmatrix} \mbox{ and } \lambda\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}\lambda x\\\lambda y\end{pmatrix}.$$ However, it is not immediately clear if there's a natural way to multiply two such coordinates, what would $$\begin{pmatrix}x\\y\end{pmatrix}\cdot \begin{pmatrix}z\\t\end{pmatrix}=?$$ be? So it seems that this space has a vector space structure, but there doesn't seem to be a natural multiplication that behaves nicely w.r.t. the addition that we already have.

I hope I somewhat convinced you already that vector spaces are fairly easy and common structures. The word linear refers to linear combinations and really summarizes what a vector space is.

Consider a function $L\colon V\to W$ from a vector space $V$ to a vector $W$. So both the domain as the target are vector spaces. Hence if $v_1,v_2\in V$ and $\lambda,\mu \in \mathbb{R}$, the expression $\lambda v_1+\mu v_2$ in $V$ is meaningful. If $L$ is just any function, the added structure of the domain and target don't give any useful information. However, we call $L$ a linear function if $$L(\lambda v_1+\mu v_2)=\lambda L(v_1)+\mu L(v_2)$$ for all $v_1,v_2$ and $\lambda,\mu$. The slogan is $L$ preserves linear combinations.

Why do we care about such functions? The answer is manifold.

• Many real life problems can be modelled with such a function (like, there are ridiculously many interesting and important examples!);
• These are easy functions and you don't need a lot of information to know the function everywhere (if you know two points on a line, you the line, similarly, if you know $L$ on enough vectors, you know $L$ everywhere).
• Linear maps between finite-dimensional vector spaces can always be modeled as multiplication by a matrix (which is extremely easy computationally).
• Non-linear problems are often way to difficult to solve explicitly, however, ofte these problems can be approximated by multiple linear problems which can be solved quickly.

Here's a more abstract example of a linear map. Consider the vector space $C^1(\mathbb{R})$ of (continuously) differentiable functions on $\mathbb{R}$ and let $C(\mathbb{R})$ be the space of continuous functions. The map $$\frac{\mathbb{d}}{\mathbb{d}x}\colon C^1(\mathbb{R})\to C(\mathbb{R}): f\mapsto \frac{\mathbb{d}f}{\mathbb{d}x}$$ which maps a function $f$ to its derivative $\frac{\mathbb{d}f}{\mathbb{d}x}$ is a linear map. The fact that is linear, expresses that differentation behaves nicely with respect to summation of functions.

Now for this last example, it doesn't provide you with new insights about differentation, nor will standard techniques of linear algebra tell you something about differentation that you didn't know already. Linear algebra really shines in situations where the linear map is a linear map between finite-dimensional vector spaces. For those maps, the techniques in linear algebra give you a lot of insight into the map itself, and as the above example shows, linear maps can be very important!