Would someone please explain what the KERNEL and SPAN of a matrix are in plain english.
that would be fantastic!
Stop thinking of a matrix as a matrix and start thinking of it as a linear mapping between two vector spaces, $\displaystyle M: U \rightarrow V$. Then the kernel of your linear map (matrix) is everything that is mapped to the zero vector, $\displaystyle \{u: uM=0 \in V\}$.
The span, on the other hand, is the image of your linear map. It is everything in V which is mapped to, $\displaystyle \{v \mid v=uM \text{ for some } u \in U\}$.
Strictly speaking, there is no such thing as the "span" of a matrix. "Span" only applies to a set of vectors. The span of a set of vectors is the set of all possible linear combinations of those vectors. The "span" of a matrix is really the span of the columns of the vector thought of as vectors.
For example, the "span" of the matrix $\displaystyle \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$ is the span of {<1, 3>, <2, 4>}. And since those are independent vectors, the span is all of $\displaystyle R^2$.
The "span" of the matrix $\displaystyle \begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix}$ is the span of {<1, 2>, <2, 4>} which, because <2, 4>= 2<1, 2>, is just "all multiples of <1, 2>".
Now suppose <x, y> is any 2-vector. Then $\displaystyle \begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}x+ 2y \\ 2x+ 4y\end{bmatrix}$. Notice that 2x+ 4y= 2(x+ 2y) so that any vector of the form Av, for this matrix A, lies in the "span" of A.
That means that another way to think of the "span" of a matrix A is as the "range" of the function f(x)= Ax.
And, of course, the kernel is the set of x such that f(x)= Ax= 0.
A more common name for the "span" of a matrix is the "column space" or "image" of the matrix.