Finding the Image of a Linear Transformation? Just a simple question, but I had a hard time finding a decent explanation. I'm confused about finding the image when given a matrix and what exactly this represents. 
For example the matrix A = \begin{bmatrix}1&2&3\\1&3&4\\1&4&5\end{bmatrix}
What is the image of $F_A$?

The image is the set of all vectors of the form $Av$ for an arbitrary vector $v$.
For example for the standard basis vector $e_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$, we know $Ae_1=\begin{bmatrix}1\\1\\1\end{bmatrix}$ is in the image.
Hint 1: what happens for the other standard basis vectors $e_2$ and $e_3$?
Hint 2: if you know $Ae_1$, $Ae_2$, and $Ae_3$, then for any $v = c_1e_1+c_2e_2+c_3e_3$ you can write $Av = c_1(Ae_1)+c_2(Ae_2)+c_3(Ae_3)$. Can you use this to describe the image of $A$ neatly?
