• Matrix Multiplication: The meaning of $Ax$ (range of $A$) and $Ax=b$.

Most students, after finishing the course linear algebra, may don‘t understand the matrix multiplication yet. Here I will show the readers, roughly speaking, the real and the most significant meaning of matrix multiplication.

Denotes a matrix $A$ in it‘s columns by $\left[ {\begin{array}{*{20}{c}}
{{A_1}}&{{A_2}}& \cdots &{{A_n}}
\end{array}} \right]$, where ${A_i}$ is the ith column of $A$. A vector $x = {\left[ {\begin{array}{*{20}{c}}
{{x_1}}&{{x_2}}& \cdots &{{x_n}}
\end{array}} \right]^T}$, where ${x_i} \in R$. Then \[Ax = \left[ {\begin{array}{*{20}{c}}
{{A_1}}&{{A_2}}& \cdots &{{A_n}}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{x_1}}\\
{{x_2}}\\
 \vdots \\
{{x_n}}
\end{array}} \right] = {x_1}{A_1} + {x_2}{A_2} +  \cdots  + {x_n}{A_n} = \sum\limits_{i = 1}^n {{x_i}{A_i}} \]

which implies that the result of $Ax$ is a linear combination of the columns of $A$ with corresponding coefficients, the components of the vector $x$. That evidently indicate that the range of $A$, $\left\{ {v:v = Ax,x \in {R^n}} \right\} \buildrel \Delta \over = {\rm{range}}\left( A \right)$, is spanned by the columns of $A$. That is, if $v \in {\rm{range}}\left( A \right)$, then $v = {a_1}{A_1} + {a_2}{A_2} +  \cdots  + {a_n}{A_n}$ for some ${a_i}$. This may be the most significant for matrix multiplication.

　　1. Simplify computation.

Suppose $A = \left[ {\begin{array}{*{20}{c}}
1&2\\
{ - 1}&4\\
0&7
\end{array}} \right]$ and $x = \left[ {\begin{array}{*{20}{c}}
2\\
1
\end{array}} \right]$. Find $Ax$.

Ans. If we use the original algorithm, we cann‘t get the result until 6 times computation. However, if we use the algorithm above, we may complete the computation whthin 3 steps, \[Ax = \left[ {\begin{array}{*{20}{c}}
1&2\\
{ - 1}&4\\
0&7
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
2\\
1
\end{array}} \right] = 2\left[ {\begin{array}{*{20}{c}}
1\\
{ - 1}\\
0
\end{array}} \right] + 1\left[ {\begin{array}{*{20}{c}}
2\\
4\\
7
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
2\\
{ - 2}\\
0
\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
2\\
4\\
7
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
4\\
2\\
7
\end{array}} \right].\]

　　2.