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$$a\cdot x_1+b\cdot x_2+c\cdot x_3+...+qx_n=\text{constant}$$ is called a linear equation because it represents the equation of a line in an $n$ dimensional space. So "linear" comes from the word "line". Basically there should not be any higher power of $x$ failing which the graph of the function will not be a straight line.

similarly

$$a(x)y+b(x)y'+c(x)y''+d(x)y'''+...+q(x)=0$$ is also called linear differential equation because all the derivatives have power=1 which is similar to the above definition of a linear equation.

A function f is called linear if: $$f(x+y)=f(x)+f(y)$$ and $$f(c\cdot x)=c\cdot f(x).$$ Here c is a constant. In this definition of linearity of function $f$ what does the word linear means? How does it relate to a straight line?

Finally what does the term linear means in case of linear vector spaces? Where is the reference to a straight line?

So, whether linear is just a word used in different contexts? Does it have different meaning in different situation? Or linearity refers to some relation to a straight line?

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    $\begingroup$ Linear vector space means that any element can be uniquely generated as linear combination of some basis elements. $\endgroup$ – gented May 15 '19 at 12:44
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    $\begingroup$ Slightly off topic, but is there a nonlinear vector space? I thought the name "linear space" is the same as "vector space", so isn't "linear vector space" redundant? $\endgroup$ – ErickShock May 15 '19 at 13:24
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    $\begingroup$ Your first equation is not a line for $n>2$. $\endgroup$ – G. Smith May 15 '19 at 16:24
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    $\begingroup$ The term linear algebras goes back to the 1880s (also here), and vectors fairly quickly entered the scene around 1900 ("quickly" in the sense of usage in early 1890s vs. usage by early 1920s), and so I suspect when axioms for abstract vector spaces started being introduced (probably mostly got "off the ground" in 1920s, but I'm just guessing now), the term "linear" got included when "space" was included. $\endgroup$ – Dave L. Renfro May 16 '19 at 13:11
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    $\begingroup$ Possible duplicate of What is the meaning of the term "linear" $\endgroup$ – YuiTo Cheng May 16 '19 at 14:04
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As you pointed out, linearity is the property of maps $f$ that satisfy $$ f(a x + y) = a f(x) + f(y)$$ for scalars $a$.

A vector space has a linear structure, in that if $x, y \in V$ then $ ax + y \in V,$ and the structure-preserving maps between vector spaces are the linear maps.

A linear map $f$ between (finite-dimensional) vector spaces can always be represented by a matrix $A$, i.e., $$ f(x) = Ax.$$

Also note that strictly speaking, the equation of a line $ f(x) = ax + b$ is not linear unless $b=0$, because a linear map must preserve the origin, i.e. map zero to zero. Only lines passing through the origin qualify as linear. General lines are examples of affine maps.

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    $\begingroup$ Your explanation is very appealing to me. Would you kindly please elaborate "A linear map 𝑓 between (finite-dimensional) vector spaces can always be represented by a matrix 𝐴, i.e., 𝑓(𝑥)=𝐴𝑥." with an example. I dont understand why you used the term "finite-dimensional". Further more a vector space can not only contain nxn matrices or row or column vectors but also functions: for example the periodic functions within x=0 to L also form a vector space. In that case, How your "A" can be a matrix? $\endgroup$ – user103515 May 17 '19 at 5:10
  • $\begingroup$ @user103515 in finite dimensions, a linear map is determined by its action on a basis, and this is precisely the information that a matrix encodes: each column of the matrix determines how each basis vector in the given ordered basis is transformed. For infinite-dimensional vector spaces, this can't work anymore since a matrix by definition has only a finite number of columns. Your example of the function space between x=0 and L is an infinite-dimensional vector space. $\endgroup$ – Nasos Evangelou-Oost May 17 '19 at 5:16
  • $\begingroup$ :So is it true that function spaces can not be finite dimensional? $\endgroup$ – user103515 May 17 '19 at 6:00
  • $\begingroup$ @user103515 no, function spaces could be either finite or infinite dimensional. The vector space of degree $\le n$ polynomial functions has as basis $\{1,x,\ldots,x^n\}$, and is therefore finite-dimensional. The vector space of all polynomial functions has a basis $\{1,x,\ldots\}$ and is infinite-dimensional. $\endgroup$ – Nasos Evangelou-Oost May 17 '19 at 6:07
  • $\begingroup$ So if i understand it properly; "A" in case of a degree n polynomial vector space can easily be a nxn matrix..... $\endgroup$ – user103515 May 17 '19 at 6:15

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