So for a linear map, it has to go through the origin(Eg.(0,0) in R^2)? Previously we mentioned that a linear map's origin has to be fixed, does that mean f(0) = 0?
wanshenl
If f is a linear map, then f(0) = 0. You can refer to 3blue1brown for a nice geometric understanding of what a linear map does. https://youtu.be/kYB8IZa5AuE?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&t=169
wanshenl
Algebraically, a linear map requires that f(u+v) = f(u) + f(v) for vectors u and v. So if the origin was not fixed, i.e. f(0) != 0, we have that f(u+0) = f(u) and f(u+0) = f(u) + f(0), and the two are not the same since f(0) != 0.
So for a linear map, it has to go through the origin(Eg.(0,0) in R^2)? Previously we mentioned that a linear map's origin has to be fixed, does that mean f(0) = 0?
If f is a linear map, then f(0) = 0. You can refer to 3blue1brown for a nice geometric understanding of what a linear map does. https://youtu.be/kYB8IZa5AuE?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&t=169
Algebraically, a linear map requires that f(u+v) = f(u) + f(v) for vectors u and v. So if the origin was not fixed, i.e. f(0) != 0, we have that f(u+0) = f(u) and f(u+0) = f(u) + f(0), and the two are not the same since f(0) != 0.
Nice explanation!