linear mapping question

[Noodles141]

Ok this concept isn't sticking for me and i can only apply it to very simple problems. The question asked is "what is the image of the circle x^2+y^2=1 under the linear map represented by the matrix (a 0,0 b)?" <----{a 0 is the top layer of the matrix and 0 b is the bottom layer}.
i get that its a scale by a and b but not to sure why just something i know. What i did was take the function as being represented as f(x,y) and and used matrix multiplication to get f(ax,by) then applied it to the formula and got a^2.x^2 + b^2.y^2=1. I'm very confused by this cause i thought we were scaling it by a factor of a and b and this is doing the opposite of what i thought it would be and i don't believe my answer is right or either i'm missing something out and my radius for my answer (that being 1) is incorrect. Would appreciate a little help and thank you in advance for taking the time to read this

 

[dougy83]

The reason the a is scaling the x and the b is scaling the y is because the linear transform matrix is multiplied by every [x; y] point thus:
[LATEX]\begin{bmatrix}
a & 0 \\
0 & b
\end{bmatrix}\begin{bmatrix}
x_i\\
y_i
\end{bmatrix}=\begin{bmatrix}
a \cdot x_i\\
b \cdot y_i
\end{bmatrix}[/LATEX]

The reason the scale appears to be working in the opposite manner can perhaps be seen from this simpler equation (where b = a):

[LATEX]a \cdot x^2 + a \cdot y^2 = 1[/LATEX]
[LATEX]\therefore x^2 + y^2 = \frac{1}{a}[/LATEX]

You can see that a larger scale factor decreases the radius.

 

[Noodles141]

ok that makes sense now haha. but is it not a^2 when put back into the formula (for the specific example u used) after scaling? if not i don't understand why it wouldn't be squared.

 

[dougy83]

I think you're correct. If you follow https://www.electro-tech-online.com/custompdfs/2013/04/PlanarEllipses.pdf through, you'll find the a^2 and b^2 gets carried through, such that [LATEX]\frac{u^2}{a^2} + \frac{v^2}{b^2} = 1[/LATEX]

 

[Noodles141]

@dougy83
Ya that's right. Talked to my clever friend haha. He said you use the jacobian of the transform matrix and that gives you the scale factor for the area under the transformation. basically the end result will be (x^2)/a +(y^2)/b=1 the area of this is equal to that of the original formula's area multiplied by the scale factor which in this case is |ab|