[Question 2015s01] Fudan Advanced Algebra II (Level 14) weekly (First teaching week)

[Problem 2015s01] set \ (M_n (\mathbb{r}) \) is the real linear space of the entire formation of the \ (n\) Order matrix, \ (\varphi\) is a linear transformation on \ (M_n (\mathbb{r}) \), so that for a given \ (A, B\in m_n (\mathbb{r}), or \ (\varphi (AB) =\varphi (a) \varphi (b) \), or \ (\varphi (AB) =\varphi (b) \varphi (a) \) is established. Proof: either \ (\varphi (AB) =\varphi (a) \varphi (b) \) is true for any of the \ (A,b\in m_n (\mathbb{r}) \), or \ (\varphi (AB) =\varphi (b) \varphi (a) \) is set for arbitrary \ (a,b\in m_n (\mathbb{r}) \).

The hints can be proved by the properties of linear space (Fudan high scrivener P164 Exercise 11) or the integration of multivariate polynomial (Fudan high scrivener P242 proposition 5.8.1).

[Question 2015s01] Fudan Advanced Algebra II (Level 14) weekly (First teaching week)