I've noticed that in Statistics, when the topic of "linearity" comes up, what is usually meant is that a fit can be made with a function of the form R ↦ R : y ( x ) = m x + qNow in linear algebra this is obviously not true (equations of the form y=mx+q are not necessarily linear).Am i mistaking something or is this just a point where statisticans and mathematicians use the same words for different things?

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I&#039;ve noticed that in Statistics, when the topic of &quot;linearity&quot; comes up, what is usually meant is that a fit can be made with a function of the form      R    ↦      R    :  y  (  x  )  =  m  x  +  qNow in linear algebra this is obviously not true (equations of the form y=mx+q are not necessarily linear).Am i mistaking something or is this just a point where statisticans and mathematicians use the same words for different things?

Josie Stephenson · Accepted Answer

It is true that in the context of linear algebra, the function   f  (  x  )  =  m  x  +  q is not considered to be &quot;linear&quot; (except in the case that   q  =  0). Instead, such a function would be called affine.However, the equation   y  =  m  x  +  q is considered a &quot;linear equation&quot; in the context of linear algebra. An equation is considered linear if it can be written in the form   f  (  x  )  =  b for some linear function   x. In this case, we could take   f  (  x  )  =  m  x and   b  =  y  −  q.

misurrosne · Accepted Answer

It is not the case that &quot;linear&quot; means only one thing in statistics.Say you have data points             (        (      x    i    ,      y    i    )  :  i  =  1  ,  …  ,  n            )       and you find the values             α      ^        ,            β      ^       for which the sum of squares of residuals       ∑          i      =      1        n              (              y        i            −                                                  y                                      ^                          i            )        2    =      ∑          i      =      1        n              (              y        i            −      (                        α          ^                    +                        β          ^                            x        i            )      )        2    ,, is as small as possible.Then the pair       (                  α        ^              ,                  β        ^              )   depends on the vector   (      y    1    ,  …  ,      y    n    ) in a linear way and depends on   (      x    1    ,  …  ,      x    n    ) in a nonlinear way, and that is exactly the meaning the term has in linear algebra. And that is why fitting a polynomial is considered linear regression just as much as fitting a straight line is linear regression. It is not because one is fitting a line that it is called linear.However, one says that correlation is a measure of &quot;linear&quot; dependence between the two variables, and there the terms &quot;linear&quot; and &quot;linear dependence&quot; do not mean the same thing they mean in linear algebra, and in this case it is about fitting a line.

I've noticed that in Statistics, when the topic of "linearity" comes up, what is usually meant is th

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