# MIT 18.06 : Linear Algebra

Row PictureColumn Picture
$\left[\begin{array}{cc}U1& V1\\ U2& V2\end{array}\right]$ $\left[\begin{array}{c}x\\ y\end{array}\right]$ = $\left[\begin{array}{c}U1*x + V1*y\\ U2*x + V2*y\end{array}\right]$ $\left[\begin{array}{cc}U1& V1\\ U2& V2\end{array}\right]$ $\left[\begin{array}{c}x\\ y\end{array}\right]$ = x $\left[\begin{array}{c}U1\\ U2\end{array}\right]$ + y $\left[\begin{array}{c}V1\\ V2\end{array}\right]$
By RowsBy Columns
Row x Matrix = RowMatrix x Column = Column
$\left[\begin{array}{cc}x& y\end{array}\right]$ $\left[\begin{array}{cc}U1& U2\\ V1& V2\end{array}\right]$ = x $\left[\begin{array}{cc}U1& U2\end{array}\right]$ + y $\left[\begin{array}{cc}V1& V2\end{array}\right]$ $\left[\begin{array}{cc}U1& V1\\ U2& V2\end{array}\right]$ $\left[\begin{array}{c}x\\ y\end{array}\right]$ = x $\left[\begin{array}{c}U1\\ U2\end{array}\right]$ + y $\left[\begin{array}{c}V1\\ V2\end{array}\right]$
BlockDotColumnsRowsColumn x Row
$\left[\begin{array}{cc}\left[B1\right]& \left[B2\right]\\ \left[B3\right]& \left[B4\right]\end{array}\right]$ $\left[\begin{array}{cc}\left[b1\right]& \left[b2\right]\\ \left[b3\right]& \left[b4\right]\end{array}\right]$ = $\left[\begin{array}{cc}\left[B1\right]\left[b1\right]+\left[B2\right]\left[b3\right]& \left[B1\right]\left[b2\right]+\left[B2\right]\left[b4\right]\\ \left[B3\right]\left[b1\right]+\left[B4\right]\left[b3\right]& \left[B3\right]\left[b2\right]+\left[B4\right]\left[b4\right]\end{array}\right]$ $\left[\begin{array}{c}- R1 -\\ - R2 -\end{array}\right]$ $\left[\begin{array}{cc}|& |\\ C1& C2\\ |& |\end{array}\right]$ = $\left[\begin{array}{cc}\left[\begin{array}{c}- R1 -\end{array}\right]\left[\begin{array}{c}|\\ C1\\ |\end{array}\right]& \left[\begin{array}{c}- R1 -\end{array}\right]\left[\begin{array}{c}|\\ C2\\ |\end{array}\right]\\ \left[\begin{array}{c}- R2 -\end{array}\right]\left[\begin{array}{c}|\\ C1\\ |\end{array}\right]& \left[\begin{array}{c}- R2 -\end{array}\right]\left[\begin{array}{c}|\\ C2\\ |\end{array}\right]\end{array}\right]$ M $\left[\begin{array}{cc}|& |\\ C1& C2\\ |& |\end{array}\right]$ = $\left[\begin{array}{cc}M\left[\begin{array}{c}|\\ C1\\ |\end{array}\right]& M\left[\begin{array}{c}|\\ C2\\ |\end{array}\right]\end{array}\right]$ $\left[\begin{array}{ccc}-& R1& -\\ -& R2& -\end{array}\right]$ M = $\left[\begin{array}{c}\left[\begin{array}{c}- R1 -\end{array}\right]M\\ \left[\begin{array}{c}- R2 -\end{array}\right]M\end{array}\right]$ $\left[\begin{array}{cc}|& |\\ C1& C2\\ |& |\end{array}\right]$ $\left[\begin{array}{c}- R1 -\\ - R2 -\end{array}\right]$ = Σ $\left[\begin{array}{c}|\\ Ci\\ |\end{array}\right]$ $\left[\begin{array}{c}- Ri -\end{array}\right]$
A=LU
 $\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right$ A
 $\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ I
 $\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right$ EA=U
 $\left|\begin{array}{cc}1& 0\\ -2& 1\end{array}\right]$ EI=E
→ L = E⁻¹ = $\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]$
 $\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]$ A
=
 $\left[\begin{array}{cc}1& 0\\ 2& 1\end{array}\right]$ L
 $\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]$ U