Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
linear_algebra:01_matrices [2014/01/20 12:25] marje |
linear_algebra:01_matrices [2014/12/14 00:03] jaan fixed spelling typos |
||
|---|---|---|---|
| Line 82: | Line 82: | ||
| <BOOKMARK:m_multiplication> | <BOOKMARK:m_multiplication> | ||
| - | ^Operaton ^Formular ^Example^ | + | ^Operation ^Formula ^Example^ |
| | Sum | $(a_{ij})+(b_{ij})=(a_{ij}+b_{ij})$ | $\begin{pmatrix}\color{blue}{1} & \color{blue}2\\\color{blue}0 &\color{blue}{-2}\end{pmatrix}+\begin{pmatrix}\color{red}{0} &\color{red}{-1}\\\color{red}1 & \color{red}2\end{pmatrix}=\begin{pmatrix}\color{blue}1+\color{red}0 & \color{blue}2+\color{red}{(-1)}\\\color{blue}0+\color{red}1 & \color{blue}{-2}+\color{red}2\end{pmatrix}=\begin{pmatrix}1 & 1\\1 & 0 \end{pmatrix}$ | | | Sum | $(a_{ij})+(b_{ij})=(a_{ij}+b_{ij})$ | $\begin{pmatrix}\color{blue}{1} & \color{blue}2\\\color{blue}0 &\color{blue}{-2}\end{pmatrix}+\begin{pmatrix}\color{red}{0} &\color{red}{-1}\\\color{red}1 & \color{red}2\end{pmatrix}=\begin{pmatrix}\color{blue}1+\color{red}0 & \color{blue}2+\color{red}{(-1)}\\\color{blue}0+\color{red}1 & \color{blue}{-2}+\color{red}2\end{pmatrix}=\begin{pmatrix}1 & 1\\1 & 0 \end{pmatrix}$ | | ||
| | Scalar multiplication | $c\cdot (a_{ij})=(c\cdot a_{ij})$ | $\color{blue}{-1}\cdot\begin{pmatrix}1 & 8 \\5 &-2 \end{pmatrix}=$ $\begin{pmatrix}\color{blue}{-1}\cdot 1 & \color{blue}{-1}\cdot 8 \\\color{blue}{-1}\cdot 5 &\color{blue}{-1}\cdot (-2)\end{pmatrix}=\begin{pmatrix}-1 & -8 \\-5 & 2 \end{pmatrix}$| | | Scalar multiplication | $c\cdot (a_{ij})=(c\cdot a_{ij})$ | $\color{blue}{-1}\cdot\begin{pmatrix}1 & 8 \\5 &-2 \end{pmatrix}=$ $\begin{pmatrix}\color{blue}{-1}\cdot 1 & \color{blue}{-1}\cdot 8 \\\color{blue}{-1}\cdot 5 &\color{blue}{-1}\cdot (-2)\end{pmatrix}=\begin{pmatrix}-1 & -8 \\-5 & 2 \end{pmatrix}$| | ||
| Line 139: | Line 139: | ||
| * $\begin{pmatrix}2 & 10 & 7\\1 & 4 & 3\end{pmatrix}\begin{pmatrix}1 & 6 & 4\\2 & 3 & 1\\2 & 1 &10\end{pmatrix}$ over $\mathbb{Z_3}$. | * $\begin{pmatrix}2 & 10 & 7\\1 & 4 & 3\end{pmatrix}\begin{pmatrix}1 & 6 & 4\\2 & 3 & 1\\2 & 1 &10\end{pmatrix}$ over $\mathbb{Z_3}$. | ||
| - | </columns> | + | </columns>\\ |
| </WRAP> | </WRAP> | ||
| Line 174: | Line 174: | ||
| Another important operation on matrices is that of taking **the transpose**. It is denoted by placing a $T$ as an exponent on the initial matrix. | Another important operation on matrices is that of taking **the transpose**. It is denoted by placing a $T$ as an exponent on the initial matrix. | ||
| - | ^Operaton ^Formular ^Example^ | + | ^Operation ^Formula ^Example^ |
| | Transpose |$(a_{ij})^T=(a_{ji})$ | $\begin{pmatrix}\color{blue}1 & \color{blue}2 & \color{blue}{-6}\\ \color{red}3 & \color{red}5 & \color{red}4\end{pmatrix}^T=\begin{pmatrix}\color{blue}1 & \color{red}3\\ \color{blue}2 & \color{red}5\\ \color{blue}{-6} & \color{red}4\end{pmatrix}$ | | | Transpose |$(a_{ij})^T=(a_{ji})$ | $\begin{pmatrix}\color{blue}1 & \color{blue}2 & \color{blue}{-6}\\ \color{red}3 & \color{red}5 & \color{red}4\end{pmatrix}^T=\begin{pmatrix}\color{blue}1 & \color{red}3\\ \color{blue}2 & \color{red}5\\ \color{blue}{-6} & \color{red}4\end{pmatrix}$ | | ||