What does it mean to say that a vector is in the column space of another?

Question

Suppose we have that a matrix $A$, and a vector $y$. When we say that $y$ is in the column space of $A$, what does that mean exactly? Does it mean we have that $y = Ax$ for some vector x and that the system is consistent? Thanks.

Answer 1

The following statements mean exactly the same thing:

• $y$ is in the column space of $A$
• If $\vec a_1,\dots,\vec a_n$ are the columns of $A$, then there are some scalars $x_1,\dots,x_n$ (that is, a column-vector $x = (x_1,\dots,x_n)$) for which $y = x_1 \vec a_1 + \cdots + x_n \vec a_n$
• For some vector $x$, we have $Ax = y$
• The system $Ax = y$ (where $y$ is the column-vector $y = (y_1,\dots,y_m)$) has a solution $x = (x_1,\dots,x_n)$ for some choice of values $x_1,\dots,x_n$
• The system $Ax = y$ is consistent

Hopefully that clears things up.