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I know that a function is called linear if it satisfies the conditions $$f(x+y)=f(x)+f(y)$$ and $$f(ax)=af(x)$$ (i.e. it preserves the properties or operation).

How is the condition for linearity given when there are multiple variable in the funciton? for ex $$f(x+y,z)$$ and $$f(ax,z)$$ I know $$f(x+y,z)=f(x,z)+f(y,z)$$ is a condition for multilinear function, but can there be multivariable function which is linear and not multilinear,ie linear in all the variables at once and not seperately (like in multilinear function). How to make sense of multivariable function that is linear and that is multilinear? Edit: Extra details.

A function $f(x,y)$ is called multlinear(bilinear to be specific) if

$$f(x+a,y)=f(x,y)+f(a,y)$$ $$f(x,y+b)=f(x,y)+f(x,b)$$ and also $$f(nx,y)=nf(x,y)=f(x,ny)$$.

So is $$f(x+a,y+b)=f(x,y+b)+f(a,y+b)+f(x+a,y)+f(x+a,b)+f(x,y)+f(a,b)+f(x,b)+f(a,y)$$

when $f(x,y)$ is multilinear and $$f(x+a,y+b)=f(x,y)+f(a,b)$$ when $f(x,y)$ is linear?

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  • $\begingroup$ Multlinear means linear in each variable separately, the classic example is $\det$. It is a weaker condition than linearity (in all variables). $\endgroup$
    – copper.hat
    Jun 30, 2018 at 5:59
  • $\begingroup$ @copper.hat I know when a multivariable function can be called a multilinear function. When can the multivariable function be called a linear function? or does being linear means defining it separately with respect to each individual variables. Also what does linearity in all variable mean? $\endgroup$
    – GRANZER
    Jun 30, 2018 at 10:45

2 Answers 2

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Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ a function. We say $f$ is linear in the $i^{th}$ variable if, given fixed $x_{j}$ for $j\neq i$ the function $T$ defined by $$T(x)=f(x_{1},x_{2},\dots,x_{i-1},x,x_{i+1},\dots,x_{n}) $$ is linear. If $f$ is linear in the $i^{th}$ variable for each $i$, we say that $f$ is multilinear.

For example, an inner product <$\cdot,\cdot$> is bilinear, because $$\left<x+y,z\right>=\left<x,z\right> + \left<y,z\right> $$ $$\left<x,y+z\right>=\left<x,y\right> + \left<x,z\right> $$

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  • $\begingroup$ @MateuRocha When would the multivariable function be called linear and not multilinear? $\endgroup$
    – GRANZER
    Jun 30, 2018 at 9:53
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    $\begingroup$ @GRANZER a function that is multilinear is linear in each variable separately. A function is linear if it's a linear in all variables simultaneously. For example the determinant function is multilinear, but not linear. $\endgroup$ Jun 30, 2018 at 11:56
  • $\begingroup$ @MateusRocha exactly what I was thinking ie a multi-variable function is linear if it is linear in all the variables simultaneously. But I don't know how to write. ie what being linear simultaneously means...what is the condition for f(x+a,y+b) to be called linear or multilinear. $\endgroup$
    – GRANZER
    Jun 30, 2018 at 12:11
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    $\begingroup$ @GRANZER Rather than think of them as variables think of it as a vector with scalar coefficients.Then you simply have to verify the usual linear condition which is $f(ax)=af(x)$ and $f(x + y)=f(x)+f(y)$ where $a$ is a scalar with $x$ and $y$ vectors. Note that scalar multiplication and vector addition impact all the variables at once which is why we say "simultaneously" for this case. $\endgroup$ Jun 30, 2018 at 21:08
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    $\begingroup$ @GRANZER for a multilinear function it should go $f(x+a,y+b) = f(x,y+b)+ f(a,y+b) = f(x,y) + f(x,b) + f(a,y) + f(a,b)$ just like you wrote and you seem to understand what being linear in all the variables means as well so I'm not sure your understanding needs correction. $\endgroup$ Jul 1, 2018 at 12:13
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If the function $\, f \,$ is the sum of linear functions in each variable separately then you can call the function linear. For example, if $\, f(x,y,z) := ax + by + cz, \,$ then $\, f(nx,ny,nz) = n \, f(x,y,z) \,$ and $$\, f(x_1\!+\!x_2, y_1\!+\!y_2, z_1\!+\!z_2) = a(x_1\!+\!x_2) \!+\! b(y_1\!+\!y_2) \!+\! c(z_1\!+\!z_2) = f(x_1,y_1,z_1) \!+\! f(x_2,y_2,z_2). \,$$

An example of a bilinear function is $\, f(x,y) = axy \,$ where $\, f(nx,y) = f(x,ny) = n\, f(x,y) \,$ and $$\, f(x_1+x_2,y_1+y_2) = f(x_1,y_1) + f(x_1,y_2) + f(x_2,y_1) + f(x_2,y_2). \,$$

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