If

$A$ is an

$m \times n$ matrix with columns

$a_1$,…,

$a_n$, and if

$x$ is in

$\Bbb{R}^n$, then the product of

$A$ and

$x$ is the linear combination of the columns in A using the corresponding entries in

$x$ as weights. In other words,

If we were to say that

$Ax=b$, then basically:

$a_1 x_1+\cdots+a_n x_n=b$
which we see b is a

**linear combination** of

$a_1,\cdots,a_n$. You will see questions where we have to solve for the entries of

$x$ again, like last section.

We say that an equation in the form of

$Ax=b$ is a

**matrix equation**.

__Properties of $Ax$__
If

$A$ is an

$m \times n$ matrix,

$u$ and

$v$ are vectors in

$\Bbb{R}^n$,

$c$ is a scalar, then:

1.

$A(u+v)=Au+Av$
2.

$A(cu)=c(Au)$