Motivation for linear transformations

Sometimes I hate classes for mathematicians. It is not their precision and formality in building the concepts, but they never give a motivation.

So in the course my professor started right with the definition of vector space, assuming I suppose that everybody knows what he is talking about. Well, reading some books for beginners like me I've realized that vector spaces are actually a generalization of working with the properties of Euclidean spaces.

Now the topic is about linear transformations. I understand the definition, they are special cases of mappings. Well, the thing is that I don't know why the definition has to be so, what is the real motivation for such a definition?. what is behind the meaning of 'linear'? My first impression is that maybe it has something to do with just preserving the operations of vectors, though I don't know why. What makes linear transformations to be special compared with those that are not linear?

Sorry for this question, maybe it is too naive but it's really important to me.

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Linear transformations arise from mapping the space to iself, for example by rotating it around a point (Or, if you want to see it another way, place an object at the origin, and move it to a new position while rotating it. The original points are related to the new points by a mapping). –  vonbrand Feb 8 '13 at 23:40
In a vector space, you can add vectors and multiply by scalars. If you then wanted to define a transformation between vector spaces that was compatible with what you could do in vector spaces, what properties might you want to have? If the transformation is called $T$, then since I can add vectors, let's say that we want $T(x+y) = T(x) + T(y)$. So the sum of the transform is the transform of the sum. We can also multiply by scalars, so let's also demand that $T(cx) = cT(x)$ for scalars. So a linear transformation $T$ basically respects anything I can possibly do to vectors. –  Zach L. Feb 8 '13 at 23:43
If it's linear, you have the superposition property. –  chaohuang Feb 9 '13 at 0:13
I have seen (forget where) the saying that mathematics is the art of linearizing problems. While this is not meant entirely seriously, there is a significant amount of truth in it. –  André Nicolas Feb 9 '13 at 0:24
@AndréNicolas You're describing engineering, not math. Pure mathematics is an art form. It does not require motivation. What is the motivation for a beautiful sunset, a baby's smile, the silent destruction of inhabited worlds? That is math. –  barrycarter Feb 9 '13 at 22:22

Vector spaces, as you correctly say, are generalizations of Euclidean spaces, in particular the structure of lines, planes, and hyper planes in Euclidean space.

Now, the definition of vector space is actually an axiomatization of some of the properties of Euclidean spaces. Now, once the abstract notion of vector space is in place one starts noticing that many things that previously did not appear to have much to do with Euclidean spaces are actually vector spaces (e.g., spaces of functions, spaces of polynomials, spaces of solutions to linear equations, spaces of solutions to certain differential equations, etc.).

With so many examples of vector spaces their study is well motivated. Now, once you have two vector spaces $V,W$ it is a very natural question to understand how they relate to each other. Since vector spaces are sets with extra structure it is most natural to consider how the sets relate when the extra structure is preserved. One way to relate sets is by function $f:V\to W$. But we don't want any old function but only those that respect the extra structure. So we demand that $f(u+v)=f(u)+f(v)$ and that $f(\alpha v)=\alpha f(v)$. And voila, there you have linear transformations motivated - you simply want to study not just a particular vector space in isolation but how different vector spaces relate.

As for calling them linear transformations, this is a bit for historical reasons. But basically, linear transformations can be shown to be precisely those that preserve all linear entities in the domain. By linear entities think of lines, planes, and hyperplanes. So such linear entities in the domain of linear transformation will be mapped to such linear entities in the codomain.

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I can list two important reasons linear transformations are important.

1. They show up everywhere. For example, the operation of taking the derivative is a linear operation the vector space of polynomials. Projections, shearings, scalings, and so on, are also examples of linear transformations.
2. They are extremely easy to describe. If you have a linear transformation $T:V \to W$, then if you know what happens to the basis vectors $\{ e_i \}$ in $V$, then you know what happens to every vector in $V$! This makes linear transformations very easy to describe: you can describe it entirely by its matrix (once you have chosen a basis for $V$ and $W$).
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I'm in engineering, not maths, but my engineering program requires a lot of math, and I heartily second both points. –  Erika Feb 9 '13 at 9:02

Agree with most answers here. However, none of those attempt to really make it explicit the importance of linear transformations, and why we really need them . Here is my attempt.

In most systems, we want to study the effect the system has on its inputs. For example, an acoustic engineer may want to study the effect of attenuation by air on sound waves. In this case, sound waves are the input and air within the propagation volume acts as a system which impacts the waves. Now, such systems are usually complex to describe. Linearization is a first approximation to describing these systems. In many cases, such approximations are justified as the nonlinear effects are not very pronounced, or may be very small for most of the input range.

Inputs can also be approximated as a superposition of linear signals (for more info, look at Fourier transform). In particular, we attempt to approximate an input signal as a superposition of certain basis input signals (preferably an orthogonal basis). You might have seen such a basis set while working with 2D vectors in physics. The i and j unit axes used to describe any 2D vector is an example of an orthogonal basis.

This helps a great deal because now we can decompose any signal into a linear combination of such basis signals. And by the property of superposition, the effect of system (linear transformation) on the input signal can be computed as a linear combination of the effect of system on basis signals. Thus, using linear system theory, we can describe the effect of any system on any signal so long as we know the effect of the system on the basis signals.

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I think I kind of elaborated on Fredrick's answer. Had not read his response before I put in mine. So I retract my statement that none try to answer why linear transformations are important. –  Nik Feb 9 '13 at 6:25

Adding to Frederik's answer, modern linear algebra evolved from trying to formalize the study of linear systems of equations $Ax=b$, which are extremely important in any quantitative science. The concept of linear transformation arises naturally in this formalism. For example, the linear system $Ax=b$ can be interpreted as $x$ passed through the linear transformation $A$ yields $b$.

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From the perspective of algebra, linear transformations are precisely the right sorts of maps between vector spaces since they preserve the operations defined on them. Two vector spaces are essentially the same if there exists a linear isomorphism between them. For Euclidean spaces, linear transformations are of geometric importance, as all isometries of the plane are affine transformations; many arguments about affine maps are easily reduced to the linear case. In analysis, generalizing the notion of a derivative to higher dimension requires defining the derivative as a special linear transformation. I don't think it would be easy to describe the language of differential forms without linear transformations. So yeah, there are lots of reasons.

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First, let's remember what linearity means. If you have a map $\underline T$, two scalars $\alpha, \beta$ and two vectors $f, g$, then

$$\underline T(\alpha f + \beta g) = \alpha \underline T(f) + \beta \underline T(g)$$

This is generally what linearity means in most contexts. Now, why do we care about linear transformations? Let's say there's a general, nonlinear map $f(x) = x'$. We can construct a linear map $\underline f_x(a) = (a \cdot \nabla) f$ at every point $x$. This is called the Jacobian, and using it allows us to determine how lengths, areas, and such are mapped by the transformation.

Linear transformations have a lot of unique properties, and because you can often construct a linear map that is related to a particular nonlinear one, we often do so. A good example is a coordinate system transformation--generally a nonlinear map. Nevertheless, if you have a vector in one of the spaces, only the Jacobian--not the full nonlinear transformation--is needed to describe hwo that vector transforms. Very convenient!

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Lots of very good reasons have been given, I'd like to add just one which is more physical/analytical : linear transformations are "nice" in the sense that the variation in the result under perturbation $f(x+\epsilon h)$ is proportional to the magnitude of the perturbation, $\epsilon$. From a physicist's point of view, that is very reassuring.

Of course lots of phenomena are not really linear, but differential calculus tells us that locally they are close to linear. If we add to this the remark 2 from Frederik Meyer, that linear mappings are a lot easier to describe than general, non-linear mappings, I'd say that it is an important motivation to study them.

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I would like to give an answer from application/engineering perspective. Sometimes following workflow works extremely well.

1. Do Linear transform to other domain.
2. Make your equivalent operation in this other domain
3. Transform back to your first domain.

Examples:

• Laplace Transformation : Solving Electrical circuits is easier in s' Laplace domain.

• Fourier Transform : Doing some image processing operations is in Fourier domain.

• Kernel Methods: This is not always a linear transformation but idea still holds and a lot of linear transformations are used. Main Applications are --- according to wikipedia --- : 3D reconstruction, bioinformatics, chemoinformatics, information extraction, text categorization, and handwriting recognition etc.

The other thing I would like to add following: We know a lot about euclidean geometry and applications at least two thousand years of knowledge. There are a lot of theorems , tricks etc exists. Since vector spaces is a generalization of euclidean spaces. When you prove a space is a vector space, you can more easily bring euclidean knowledge to this space.

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Linear transformations $T$ preserve linear relations between individual vectors: If $x+y=z$ then $T(x)+T(y)=T(z)$, and if $y=\lambda\>x$ then $T(y)=\lambda\>T(x)$.

Eliminating the variable $z$ from the first and the variable $y$ from the second property one arrives at the usual axioms for such transformations.

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