Fudan University 2016--2017 first semester (Level 16) Advanced Algebra I final exam seventh big question answer

Seven, (10 points) set $A, b$ are $m \times n$ order matrix, to meet the $A ' B+b ' a=0$. Proof: $ $r (a+b) \geq\max\{r (A), R (B) \},$$ and the necessary and sufficient conditions for the establishment of an equal sign are the existence $m the square matrix $P $, making $B =pa$ or $A =pb$.

The law of $A ' b+b ' a=0$ can be $$ (a+b) ' (a+b) =a ' a+b ' b.$$ set $V _a\subseteq\mathbb{r}^n$ as a linear equation group $Ax =0$ of the solution space, $V _b$ and $V _{a+b}$ Similarly defined, there is $V _a\cap V_b\subseteq v_{a+b}$. Conversely, for any $x _0\in v_{a+b}$, on both sides of the equation at the same time left multiply $x _0 ' $, right multiply $x _0$, can get $$0=x_0 ' (a+b) ' (a+b) x_0=x_0 ' A ' ax_0+x_0 ' B ' bx_0= (ax_0) ' (AX_0) + (bx_0) ' (BX_0), $$ thus have $Ax _0=bx_0=0$, namely $x _0\in v_a\cap v_b$, so $V _a\cap v_b=v_{a+b}$. Note that $V _{a+b}\subseteq v_a$, $V _{a+b}\subseteq v_b$, the dimension formula of the solution space can be $r (a+b) \geq R (A) $, $r (a+b) \geq R (B) $, thereby $r (A+B) \geq\max\ {R (A), R (B) \}$ established. If the equals sign is established, may wish to set up $r (a+b) =r (A) \geq R (B) $, then $V _{a+b}=v_a\subseteq v_b$, by the fourth chapter of the White Paper answer to question 13: The existence of $m $ order matrix $P $, making $B =pa$. Conversely, if $B =pa$, then $r (B) \leq R (a) $ and $r (a+b) =r ((i_m+p) a) \leq R (a) $, thereby establishing the equals sign.

Certificate of Law two by the higher garbage textbook of the third chapter of the review 41 or the White paper example 3.72 know: $ $r (a+b) =r ((a+b) ' (a+b)) =r (A ' a+b ' B). $$ noted that $A ' a$ and $B ' b$ are semi-positive matrices, so the example from the white Paper 9.73 Known: $ $r (a ' a+b ' B) =r (a ' a\mid B ' b) \geq \max\{r (a ' a), R (b ' B) \}=\max\{r (a), R (b) \}.\,\,\,\,\box$$

note that the students are completely right: Ningsheng, Zhu Minzhe, Shen Yinan, Hotauan, Dong Ji ze.

Fudan University 2016--2017 first semester (Level 16) Advanced Algebra I final exam seventh big question answer