begins - Moozonian

${\begin{aligned}\mathbf {a} \times \mathbf {b} =[\mathbf {a} ]_{\times }\mathbf {b} &={\begin{bmatrix}\,0&\!-a_{3}&\,\,a_{2}\\\,\,a_{3}&0&\!-a_{1}\\-a_{2}&\,\,a_{1}&\,0\end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\end{bmatrix}}\\\mathbf {a} \times \mathbf {b} ={[\mathbf {b} ]_{\times }}^{\mathrm {\!\!T} }\mathbf {a} &={\begin{bmatrix}\,0&\,\,b_{3}&\!-b_{2}\\-b_{3}&0&\,\,b_{1}\\\,\,b_{2}&\!-b_{1}&\,0\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}},\end{aligned}}$

Sensitive Content Hidden
Turn off Family Mode to view

en.wikipedia.org

Visit

{\displaystyle {\begin{aligned}\mathbf {a} \tim...

${\begin{bmatrix}dx\\dy\end{bmatrix}}={\begin{bmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \theta }}\\{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \theta }}\end{bmatrix}}{\begin{bmatrix}dr\\d\theta \end{bmatrix}}={\begin{bmatrix}\cos \theta &-r\sin \theta \\sin\theta &r\cos \theta \end{bmatrix}}{\begin{bmatrix}dr\\d\theta \end{bmatrix}}$

Sensitive Content Hidden
Turn off Family Mode to view

vi.wikipedia.org

Visit

Phân phối chuẩn – Wikipedia tiếng Việt

${\begin{aligned}\mathbf {I} _{1}&={\begin{bmatrix}1\end{bmatrix}},\\[4pt]\mathbf {I} _{2}&={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\\[4pt]\vdots &\\[4pt]\mathbf {I} _{n}&={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}\end{aligned}}$

Sensitive Content Hidden
Turn off Family Mode to view

en.wikipedia.org

Visit

{\displaystyle {\begin{aligned}\mathbf {I} _{1}...

${\begin{aligned}{\begin{bmatrix}{\underline {2}}&{\underline {3}}&{\underline {4}}\\1&0&0\\\end{bmatrix}}{\begin{bmatrix}0&{\underline {1000}}\\1&{\underline {100}}\\0&{\underline {10}}\\\end{bmatrix}}&={\begin{bmatrix}3&{\underline {2340}}\\0&1000\\\end{bmatrix}}.\end{aligned}}$

Sensitive Content Hidden
Turn off Family Mode to view

en.wikipedia.org

Visit

{\displaystyle {\begin{aligned}{\begin{bmatrix}...

${\begin{bmatrix}1&3&1\\1&0&0\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\end{bmatrix}}={\begin{bmatrix}1+0&3+0&1+5\\1+7&0+5&0+0\end{bmatrix}}={\begin{bmatrix}1&3&6\\8&5&0\end{bmatrix}}$

Sensitive Content Hidden
Turn off Family Mode to view

en.wikipedia.org

Visit

{\displaystyle {\begin{bmatrix}1&3&1\\1&0&0\end...

${\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}-\lambda {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &0&0\\0&3-\lambda &4\\0&4&9-\lambda \end{vmatrix}},\\[6pt]&=(2-\lambda ){\bigl [}(3-\lambda )(9-\lambda )-16{\bigr ]}=-\lambda ^{3}+14\lambda ^{2}-35\lambda +22.\end{aligned}}$

Sensitive Content Hidden
Turn off Family Mode to view

en.wikipedia.org

Visit

{\displaystyle {\begin{aligned}\det(A-\lambda I...

${\begin{aligned}(A-3I)\mathbf {v} _{\lambda =3}&={\begin{bmatrix}-1&{\hphantom {-}}1\\{\hphantom {-}}1&-1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}\\-1v_{1}+1v_{2}&=0;\\1v_{1}-1v_{2}&=0\end{aligned}}$

Sensitive Content Hidden
Turn off Family Mode to view

en.wikipedia.org

Visit

{\displaystyle {\begin{aligned}(A-3I)\mathbf {v...

${\begin{aligned}(A-I)\mathbf {v} _{\lambda =1}&={\begin{bmatrix}1&1\\1&1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}\\1v_{1}+1v_{2}&=0\end{aligned}}$

Sensitive Content Hidden
Turn off Family Mode to view

en.wikipedia.org

Visit

{\displaystyle {\begin{aligned}(A-I)\mathbf {v}...

${\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&1\\1&2\end{bmatrix}}-\lambda {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}\\[6pt]&=3-4\lambda +\lambda ^{2}\\[6pt]&=(\lambda -3)(\lambda -1).\end{aligned}}$

Sensitive Content Hidden
Turn off Family Mode to view

en.wikipedia.org

Visit

{\displaystyle {\begin{aligned}\det(A-\lambda I...

$a_{y}={\begin{bmatrix}b_{z}\\b_{x}\end{bmatrix}}\times {\begin{bmatrix}c_{z}\\c_{x}\end{bmatrix}},\ a_{z}={\begin{bmatrix}b_{x}\\b_{y}\end{bmatrix}}\times {\begin{bmatrix}c_{x}\\c_{y}\end{bmatrix}}$

Sensitive Content Hidden
Turn off Family Mode to view

en.wikipedia.org

Visit

{\displaystyle a_{y}={\begin{bmatrix}b_{z}\\b_{...