Cannot understand this property about column space

Question

Here is a property given to me in my textbook.

$ ColA=\Bbb R^m \text{ if and only if the equation } Ax=b \text{ has a solution for every } b \text{ in } \Bbb R^m $.

What does it mean by every $b$?

Answer 1

Let $a_i$ be the $i$th column of an $n\times m$ matrix $A$. Then given $x = \pmatrix{x_1 \\ x_2 \\ \vdots \\ x_m}$, one way of calculating the product $Ax$ is like $$Ax = x_1a_1 + x_2a_2 + \cdots + x_ma_m$$

Thus you can see that $Ax=b$ really means that $b$ is a linear combination of the columns of $A$. Thus $b \in \operatorname{Col}(A)$.

What that means is that if you make an equation $Ax=b$ where $b \not\in \operatorname{Col}(A)$, then the equation can not have any solution $x$.

So $\operatorname{Col}(A)$ is the whole space $\Bbb R^m$ if and only if every $b \in \Bbb R^m$ is a linear combination of the columns of $A$, i.e. if there is a solution to $Ax=b$ for every $b$.

Answer 2

$Ax$ can be viewed as a linear combination of the columns of $A$, with the $i$th coefficient given by the $i$th coordinate of $x$. Let's look at it with a $2\times 2$ matrix:

$$\begin{bmatrix} a& b \\ c& d\end{bmatrix} \begin{bmatrix} x_1 \\ x_2\end{bmatrix} = \begin{bmatrix}x_1a+x_2b \\ x_1c+x_2 d\end{bmatrix} = \begin{bmatrix}x_1a \\ x_1c\end{bmatrix}+\begin{bmatrix}x_2b\\x_2d\end{bmatrix} = x_1\begin{bmatrix}a \\ c\end{bmatrix} + x_2\begin{bmatrix} b \\d\end{bmatrix}$$

So an element $y \in \Bbb R^m$ is in the column space in the matrix (as in the right-hand side) if and only if there is some vector $x$ such that $Ax=y$ (as in the left-hand side).

This means that the column space of $A$ is the whole vector space if and only if every vector $y$ in the vector space can be written as $Ax=y$ for some $x$.

Answer 3

The coluumn space of $A$ is the range of the linear map, say $u$, associated to $A$, so

$Ax=b\;$ has solutions

means

$u(x)=b\;$ has solutions, i.e. $\;b\in\operatorname{Im}u$, or $\;b\;$ lies in the column space of $A$.