If you consider the discriminant $\Delta= b^2-4ac$ of a quadratic equation as a linear transformation, what does it requiere to prove it?
Context: Linear Algebra course.
I tried by myself to prove it, using two conditions of linear transformations: $$T(u+v)=T(u)+T(u) $$ And, $$T(\lambda u)= \lambda T( u)$$ I need to prove it.Suggestions to verify the proof? .Thanks in advance.
No. For example, multiplying each coefficient of the quadratic equation by $k\ne0$, while leaving its roots unchanged, multiplies $\Delta$ by $k^2$. This is because $\Delta$ is a homogeneous polynomial in the coefficients of degree $2$. (By contrast, $\sqrt{\Delta}$ is a homogeneous function of degree $1$, albeit not a polynomial one. In both cases, "degree" is short for degree of homogeneity.) Other transformations of the coefficients will in general have even more complicated effects on $\Delta$.