What is the geometric meaning of scaling an object by a factor $k$ along the line $y=x$? What will be the shape of a square with vertices $(2,1)$, $(3,2)$, $(3,1)$, and $(2,2)$ if it is scaled by a factor $5$ along the line $y=x$? I really appreciate your help on this!
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The property of "scaling along $x=y$ by a factor of $5$" does not determine a specific linear transformation. Therefore one cannot say what happens to the square you specify except that the corner $(2,2)$ of the square is mapped to $(10,10)$.
Hagen Knaf
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I assume you want to stretch along the line $y=x$ while leaving dimensions perpendicular to the line unchanged. This would be similar to stretching horizontally by multiplying all $x$ coordinates by $5$ while leaving the $y$ coordinates unchanged. The easiest way is to rotate axes. Define a new set of axes by $s=x+y, t=y-x$. The $s$ axis is the line $y=x$ and the $t$ axis is the line $y=-x$. We can invert the transformation, finding $x=\frac 12(s-t), y=\frac 12(s+t)$ Then to stretch along $y=x$ by a factor $5$ we change $s$ to $5s$. Now $x'=\frac 12(5s-t)=\frac 12(6x+4y), y'=\frac 12 (5s+t)=\frac 12(4x+6y)$. This leaves the origin unchanged, moves $(1,1)$ to $(5,5)$ and leaves $(1,-1)$ unchanged as requested.
Ross Millikan
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