## Multiplication $Ax$ using Columns of $A$

$A\boldsymbol{x}$ is a linear combination of columns of $A$.

Linear combination of all the columns of $A$ produce the $\textbf{column space}$ of $A$.

$b$ is in the column space of $A$ when $A\boldsymbol{x}=b$ has a solution.

A column of $A$ is $\textbf{dependent}$ if it is a linear combination of other columns of $A$ else it is $\textbf{independent}$.

### Independent columns and the rank of $A$

A $basis$ for a subspace is a full set of $independent$ vectors. All vectors in the space are linear combinations of the basis vectors.

#### Finding the basis $C$ of a column space of $A$

If column $1$ of $A$ is not all zero, put it into matrix $C$.

If column $2$ of $A$ is not a multiple of column $1$, put it into $C$.

If column $3$ of $A$ is not a combination of columns $1$ and $2$, put it into $C$. Continue.

At the end $C$ will have $r$ columns $(r \leq n)$.

They will be the $basis$ for the column space of $A$.

The number $r$ is the $\textbf{rank}$ of $A$. It is also the rank of $C$. It counts independent columns. The number $r$ is the $\text{dimension}$ of the column space of $A$ and $C$(same space).

## References

Linear Algebra and Learning from Data