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$T(av_1 + bv_2) = aT(v_1) + bT(v_2)$

Why is this called linear? $f(x) =ax + b$, the simplest linear equation does not satisfy $f(x_1 + x_2) = f(x_1) + f(x_2)$.

Thank you.

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marked as duplicate by Workaholic, jameselmore, Tom-Tom, kamil09875, hardmath Jan 27 at 15:28

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

The term linear is not in reference to the equation of a line, as in a polynomial of degree one. Linearity means that both addition and scalar multiplication are preserved: – zahbaz Jan 23 at 5:34
Formatting tips here. – probablyme Jan 23 at 5:39
Sorry, but apparently all lines of interest pass through $0$. – MPW Jan 23 at 5:45
@MPW : All (non-affine) linear subspaces of a vector space contain $0$. The affine subspaces need not do so. – Eric Towers Jan 23 at 23:05
As I think $f=\text{constant}$ is the most simplest one. – Nil Mal Feb 2 at 9:25

Yep, in high school we say $f(x) = 3x + 4$ is a linear function, but as you point out, in linear algebra it's not. It's irritating.

The word for $x \mapsto ax + b$ is affine transformation, and this is the term you'll hear elsewhere in higher math.

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As explained there, the term linear mapping was coined by Hermann Graßmann. It describes mappings which preserve the linear structure of a space, meaning the way scaling the length of a vector parameterizes a line. If you apply a linear mapping, the image will still be a line.

Now, that's actually true for affine maps as well, so it could be argued that the high school term, using linear to mean functions of the form $\backslash x \mapsto a\cdot x + b$, is actually more meaningful. But alas, sometimes sub-optimal terminology sticks. There has by now been so much written about linear maps meaning functions that fulfill $f(\mu\cdot \mathbf v + \nu\cdot \mathbf w) = \mu\cdot f(\mathbf v) + \nu\cdot f (\mathbf w)$, that it would mostly cause confusion to use it for anything else.

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Expressions of the form $av_1 + bv_2$ are called linear combinations. Linear transformations are the functions sending linear combinations to linear combinations (preserving coefficients). That is, a function is called linear when it preserves linear combinations.

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You may refer to them as "$k$-vectorspace homomorphisms" if you wish.

In a little more detail: if $k$ is a field, then a $k$-vectorspace can be thought of as a set $X$ equipped with, for each sequence $a \in k^n$, a corresponding "linear combination mapping" $X \leftarrow X^n$. By convention, the structure-preserving mappings between $k$-vectorspaces are called "$k$-linear transforms," or simply $k$-linear, probably because the phrase "linear-combination-mapping-preserving mapping" is altogether too confusing. Like I said, you may refer to them as $k$-vectorspace homomorphisms if you wish.

But this just pushes the issue back: why are they called, of all things, linear combinations? Well, if $X$ is a $k$-vectorspace we're given a single vector $x \in X$, then $$\{ax : a \in k\}$$ is the set of all elements of $X$ that can be obtained by applying linear combination mappings to $x.$ In the special case where $k=\mathbb{R}$, this can be visualized as a line through the origin.

So "linear combination" is probably best thought of as a shorthand for "line-through-the-origin combination." Of course, if we've got two vectors, then we're potentially talking about a plane through the origin. And so on.

In fact, linear algebra isn't really about lines at all; its really about flat things that pass through the origin. Perhaps it should have been called "algebra with respect to the origin" instead!

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First of all, the equation is true in the trivial case, where b=0.

As others have pointed out, when b is not equal to zero, the result is called an affine transformation.

By definition, an affine transformation does preserve the other underlying properties of the original linear function, because it is a "parallel" shift That's why it's considered a "linear" transformation, even though the term is, in fact, slightly misleading in this context.

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Linearity has different (yet equivalent) definitions. One is what you described: preservation of linear combinations (linear combination of inputs yields linear combination of outputs), which is a godsend when solving equations - solve each part separately, and you have a general solution. In physics, this is called the principle of superposition. Secondly, it's linear in the sense of a linear function (meaning first powers in all terms). Linear functions, either in scalar context ($ax+b$) or in multiple variables (linear algebra, matricess and everything that goes with it) satisfy the superposition principle, when $b=0$. However, even when $b\neq 0$ (general affine transformations), you can work in homogeneous coordinates. Just introduce another variable that assumes a value of $1$ in your expression, and you see that the affine transform is linear (in the superposition sense) in a higher dimensional space, and projected down (the projection is in this case literal - in 3D graphics, $4\times 4$ matrices are used explicitly to render everything on your screen). Not every superposition preserves the affine transform, but linear interpolations do. This is the cornerstone of the formalism used to deal with differences between absolute and relative quantities (prominent in physics). Example: represent all positions as (x,y,z,1), and displacements as (x,y,z,0): in that case, it's obvious that a difference of positions is a displacement, an interpolation of positions is a position, position + scaled displacement = position, but scaling a position makes no sense (2*London = ?).

In short: offset of general affine transforms can be factored away into an extra dimension of homogeneous space, then all definitions of linearity make sense in the same way.

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