Different perspectives of a simple matrix equation of the form
Geometric picture
Column picture
This is a more interesting and less common perspective. Think of each column of matrix
as a column vector.
![\left|\begin{matrix}<br />1 & 2 & 3\\<br />4 & 5 & 6\\<br />7 & 8 & 9<br />\end{matrix}<br /> \right|](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%5Cbegin%7Bmatrix%7D%0A1%20%26%202%20%26%203%5C%5C%0A4%20%26%205%20%26%206%5C%5C%0A7%20%26%208%20%26%209%0A%5Cend%7Bmatrix%7D%0A%20%5Cright%7C%20)
=![\left| \begin{matrix} 1\\4\\7 \end{matrix}\right|](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%20%5Cbegin%7Bmatrix%7D%201%5C%5C4%5C%5C7%20%5Cend%7Bmatrix%7D%5Cright%7C)
The solution for
, ie the values of
that will satisfy the equation, is the point where the 2 lines, 3 planes etc.. meet. This is the picture that is normally presented.
Column picture
This is a more interesting and less common perspective. Think of each column of matrix
Each of the column vectors of
are scaled by
and then added to get
. So
is the scaling factor for the first column,
is the scaling factor for second column... etc.
For example if
,
will be equal to
which is the first column of
.
Isn't that nice?
Row picture
Better example to see what is going on here is to look at![\left|\begin{matrix}<br />x_1 & x_2&x_3<br />\end{matrix}\right| <br />\left|\begin{matrix}<br />1&2&3\\<br />3&4&5\\<br />7&8&9<br />\end{matrix}\right|=<br />\left|\begin{matrix}<br />b_1 & b_2&b_3<br />\end{matrix}\right|](https://www.google.com/chart?cht=tx&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cleft%7C%5Cbegin%7Bmatrix%7D%0Ax_1%20%26%20x_2%26x_3%0A%5Cend%7Bmatrix%7D%5Cright%7C%20%0A%5Cleft%7C%5Cbegin%7Bmatrix%7D%0A1%262%263%5C%5C%0A3%264%265%5C%5C%0A7%268%269%0A%5Cend%7Bmatrix%7D%5Cright%7C%3D%0A%5Cleft%7C%5Cbegin%7Bmatrix%7D%0Ab_1%20%26%20b_2%26b_3%0A%5Cend%7Bmatrix%7D%5Cright%7C)
Better example to see what is going on here is to look at
Here
is a scaling factor of the first row,
for the second row etc.
For example if
,
will be
which is the first row of
.