Today I started my linear algebra class by motivating the determinant using a little bit of history and applications. In particular I talked about determinant appearing in the change of variables in integration of multivariate functions in calculus:Then after the class I noticed that my picture of what happens when we linearize the image wasn’t very clear as I had drawn it. So I decided to take a look at an actual example. The following code takes a function that maps to and draws three regions:
- the unit cube in blue,
- the image of the unit cube under in red, and
- the linearization of the image by Jacobian of at the origin
def show_linearization(f,n=100): # f is a function that maps R^2 to R^2 for example # f(x,y) = ((x^2+2*(x+2)*(y+1))/4-1,y^2-(x+1)*y) # The output is three areas: # the unit cube # the image of unit cube under f # the linearization of the image using the Jacobian of f at (0,0) var('x y') A = jacobian(f,(x,y))(0,0) p = f(0,0) domxy =  imgxy =  jacxy =  for i in range(n+1): for j in range(n+1): domxy.append((i/n,j/n)) imgxy.append(f(i/n,j/n)) jacxy.append(p+A*vector([i/n,j/n])) P = points(domxy,color="blue",aspect_ratio=1, alpha = .3) Q = points(imgxy,color="red",aspect_ratio=1, alpha = .3) R = points(jacxy,color="green",aspect_ratio=1, alpha = .3) (P+Q+R).show()
Here is a sample run:
f(x,y) = ((x^2+2*(x+2)*(y+1))/4-1,y^2-(x+1)*y) show_linearization(f)
and the output is:
I chose the function in a way that for simplicity. Here it is on sage cell server for you to play around with it: link.
What do you think? Any suggestions or comments?