- [[linear combination of rows]], [[linear combinations]], [[linear regression closed-form matrix solution]]
# Idea
Multiply a matrix $A$ with a column vector $x$ is equivalent to taking the linear combination of the columns of $A$, or the [[dot product]] between the rows of $A$ and columns of $x$. This operation is a **column operation**.
A matrix times a column gives us a column.
![[s20220725_010905.png]]
![[s20220725_011218.png]]
## Example 1
If we post-multiply a matrix $A$ with the **column vector** $x$, we're computing the linear combination of the columns in matrix $A$.
$Ax= b$
$
\left[\begin{array}{cc}
2 & -1 \\
-1 & 2
\end{array}\right]\left[\begin{array}{l}
x \\
y
\end{array}\right]=\left[\begin{array}{l}
0 \\
3
\end{array}\right]
$
$
x\left[\begin{array}{c}
2 \\
-1
\end{array}\right]+y\left[\begin{array}{c}
-1 \\
2
\end{array}\right]=\left[\begin{array}{l}
0 \\
3
\end{array}\right]
$
The elements in vector/matrix $x$ are [[scalars]]. They linearly scale the columns in matrix $A$.
![[s20220725_012954.png]]
![[s20220725_005743.png]]
# References
- [Lec 1 | MIT 18.06 Linear Algebra, Spring 2005 - YouTube](https://youtu.be/ZK3O402wf1c?t=565)
- [3Blue1Brown - Matrix multiplication as composition](https://www.3blue1brown.com/lessons/matrix-multiplication)
- [3Blue1Brown - Matrix multiplication as composition](https://www.3blue1brown.com/lessons/matrix-multiplication#title)
- [2. Elimination with Matrices. - YouTube](https://youtu.be/QVKj3LADCnA?t=1290)