高等代数: 多项式与线性代数

\begin{equation*} A(B+C)=AB+AC=0,\ A(kB)=k(AB)=0, \end{equation*}

\begin{equation*} \begin{array}{c} B_{11}=(\xi_1,0,\cdots ,0),B_{12}=(0,\xi_1,0,\cdots ,0),\cdots ,B_{1n}=(0,\cdots ,0,\xi_1),\\ B_{21}=(\xi_2,0,\cdots ,0),B_{22}=(0,\xi_2,0,\cdots ,0),\cdots ,B_{1n}=(0,\cdots ,0,\xi_2),\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ B_{n-r,1}=(\xi_{n-r},0,\cdots ,0),B_{n-r,2}=(0,\xi_{n-r},0,\cdots ,0),\cdots ,B_{n-r,n}=(0,\cdots ,0,\xi_{n-r}), \end{array} \end{equation*}

\begin{equation*} (\sum\limits_{i=1}^{n-r} k_{i1}\xi_i,\sum\limits_{i=1}^{n-r} k_{i2}\xi_i,\cdots ,\sum\limits_{i=1}^{n-r} k_{in}\xi_i)=0, \end{equation*}

\begin{equation*} \sum\limits_{i=1}^{n-r} k_{i1}\xi_i=\sum\limits_{i=1}^{n-r} k_{i2}\xi_i=\cdots =\sum\limits_{i=1}^{n-r} k_{in}\xi_i=0. \end{equation*}

\begin{equation*} B_{11},B_{12},\cdots ,B_{1n},B_{21},B_{22},\cdots ,B_{2n},\cdots ,B_{n-r,1},B_{n-r,2},\cdots ,B_{n-r,n} \end{equation*}

\begin{equation*} A\beta_j=0,\ \forall 1\leq j\leq n, \end{equation*}

\begin{equation*} B=(\sum\limits_{i=1}^{n-r} a_{i1}\xi_i,\sum\limits_{i=1}^{n-r} a_{i2}\xi_i,\cdots ,\sum\limits_{i=1}^{n-r} a_{in}\xi_i)=\sum\limits_{i=1}^{n-r}\sum\limits_{j=1}^n a_{ij}B_{ij}, \end{equation*}

\begin{equation*} \alpha=(a_1,a_2,\cdots ,a_n)^T,\beta=(b_1,b_2,\cdots ,b_n)^T\in U, \end{equation*}

\begin{equation*} \alpha-\frac{a_1}{b_1}\beta=(0,a_2-\frac{a_1b_2}{b_1},\cdots ,a_n-\frac{a_1b_n}{b_1})^T\in U. \end{equation*}

\begin{equation*} k_1Y_1+k_2Y_2=k_1(AX_1)+k_2(AX_2)=A(k_1X_1+k_2X_2)\in V. \end{equation*}

\begin{equation*} \begin{array}{rcl} AX &= &(A_1,A_2,\cdots ,A_n)\begin{pmatrix}x_1\\x_2\\ \vdots\\x_n\end{pmatrix}\\ &= &x_1A_1+x_2A_2+\cdots +x_nA_n\\ &\in &\langle A_1,A_2,\cdots ,A_n\rangle,\end{array} \end{equation*}

\begin{equation*} V_1+V_2=\langle\alpha_1,\alpha_2 \rangle +\langle\beta_1,\beta_2\rangle=\langle\alpha_1,\alpha_2 ,\beta_1,\beta_2\rangle , \end{equation*}

\begin{equation*} A=(\alpha_1,\alpha_2,\beta_1,\beta_2)=\begin{pmatrix} 1 & -1 & 2 & 1\\ 2 & 1 & -1 & -1\\ 1 & 1 & 0 & 3\\ 0 & 1 & 1 & 7 \end{pmatrix}\xrightarrow{rref}\begin{pmatrix} 1 & 0 & 0 & -1\\ 0 & 1 & 0 & 4\\ 0 & 0 & 1 & 3\\ 0 & 0 & 0 & 0 \end{pmatrix}, \end{equation*}

\begin{equation*} \dim (V_1\cap V_2)=\dim V_1+\dim V_2-\dim (V_1+V_2)=2+2-3=1. \end{equation*}

\begin{equation*} \alpha=x_1\alpha_1+x_2\alpha_2=x_3\beta_1+x_4\beta_2, \end{equation*}

\begin{equation*} V_1+V_2=\langle\alpha_1,\alpha_2,\alpha_3 \rangle +\langle\beta_1,\beta_2\rangle=\langle\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2\rangle , \end{equation*}

\begin{equation*} X_1=(1,2,-1,-2)^T,X_2=(3,1,1,1)^T,X_3=(-1,0,1,-1)^T, \end{equation*}

\begin{equation*} Y_1=(2,5,-6,-5)^T,Y_2=(-1,2,-7,3)^T, \end{equation*}

\begin{equation*} A=\begin{pmatrix} 1&3&-1&2&-1\\ 2&1&0&5&2\\ -1&1&1&-6&-7\\ -2&1&-1&-5&3 \end{pmatrix}\xrightarrow{rref}\begin{pmatrix} 1&0&0&3&0\\ 0&1&0&-1&0\\ 0&0&1&-2&0\\ 0&0&0&0&1 \end{pmatrix}, \end{equation*}

\begin{equation*} \dim (V_1\cap V_2)=\dim V_1+\dim V_2-\dim (V_1+V_2)=3+2-4=1. \end{equation*}

\begin{equation*} \alpha=x_1\alpha_1+x_2\alpha_2+x_3\alpha_3=x_4\beta_1+x_5\beta_2, \end{equation*}

\begin{equation*} x_1\alpha_1+x_2\alpha_2+x_3\alpha_3-x_4\beta_1-x_5\beta_2=0, \end{equation*}

\begin{equation*} x_1X_1+x_2X_2+x_3X_3-x_4Y_1-x_5Y_2=0, \end{equation*}

\begin{equation*} \dim \left(\sum\limits_{i=1}^{s-1} V_i\right)=\sum\limits_{i=1}^{s-1} \dim V_i-\sum\limits_{k=2}^{s-1}\dim\left[(\sum\limits_{j=1}^{k-1}V_j)\cap V_k\right]. \end{equation*}

\begin{equation*} \dim\left(\sum\limits_{i=1}^{s} V_i\right)=\dim\left(\sum\limits_{i=1}^{s-1} V_i\right)+\dim V_s-\dim\left[\left(\sum\limits_{i=1}^{s-1} V_i\right)\cap V_s\right]. \end{equation*}

\begin{equation*} \dim \left(\sum\limits_{i=1}^{s} V_i\right)=\sum\limits_{i=1}^{s-1} \dim V_i-\sum\limits_{k=2}^{s-1}\dim\left[(\sum\limits_{j=1}^{k-1}V_j)\cap V_k\right]+\dim V_s-\dim\left[\left(\sum\limits_{i=1}^{s-1} V_i\right)\cap V_s\right]. \end{equation*}

\begin{equation*} \dim \left(\sum\limits_{i=1}^{s} V_i\right)=\sum\limits_{i=1}^{s} \dim V_i-\sum\limits_{k=2}^{s}\dim\left[(\sum\limits_{j=1}^{k-1}V_j)\cap V_k\right]. \end{equation*}

\begin{equation*} A_1+A_2=\begin{pmatrix} x_1+x_2&y_1+y_2\\y_1+y_2&0 \end{pmatrix}\in U,\ cA_1=\begin{pmatrix} cx_1&cy_1\\cy_1&0 \end{pmatrix}\in U, \end{equation*}

\begin{equation*} B_1=\begin{pmatrix} b_{11}-b_{22}&b_{21}\\b_{21}&0 \end{pmatrix}\in U,B_2=\begin{pmatrix} b_{22}&b_{12}-b_{21}\\0&b_{22} \end{pmatrix}\in W, \end{equation*}

\begin{equation*} tr(A+B)=tr(A)+tr(B)=0,\ tr(kA)=k\cdot tr(A)=0, \end{equation*}

\begin{equation*} 0=tr(A)=tr(aE_n)=na. \end{equation*}

\begin{equation*} tr(B-aE_n)=tr(B)-na=0, \end{equation*}

\begin{equation*} C=B-aE_n\in U,D=aE_n\in W, \end{equation*}

\begin{equation*} 0=tr(A)=tr(aE_n)=na. \end{equation*}

\begin{equation*} \sum\limits_{1\leq i\neq j\leq n}a_{ij}E_{ij}+\sum\limits_{k=1}^{n-1}a_{kk}(E_{kk}-E_{nn})=0, \end{equation*}

\begin{equation*} \begin{pmatrix} a_{11}&\cdots&a_{1,n-1}&a_{1n}\\ a_{21}&\cdots&a_{2,n-1}&a_{2n}\\ \vdots&&\vdots&\vdots\\ a_{n-1,1}&\cdots&a_{n-1,n-1}&a_{n-1,n}\\ a_{n1}&\cdots&a_{n,n-1}&-\sum\limits_{k=1}^{n-1}a_{kk} \end{pmatrix}=0, \end{equation*}

\begin{equation*} A=\sum\limits_{1\leq i\neq j\leq n}a_{ij}E_{ij}+\sum\limits_{k=1}^{n-1}a_{kk}(E_{kk}-E_{nn}), \end{equation*}

\begin{equation*} \dim V_1'=\dim V_2'=m+1. \end{equation*}

\begin{equation*} V=V_1\oplus W=V_2\oplus W. \end{equation*}