TITLE: Why is kernel important?
QUESTION [9 upvotes]: Why are we interested in looking at the kernel and range (image) of a linear transformation on a linear algebra course?

REPLY [6 votes]: One example to keep in mind is a projection $\pi: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ onto a plane $\Pi$ in $\mathbb{R}^3$.  This type of linear transformation comes up all the time in computer graphics, where you are trying to represent something three-dimensional on a two-dimensional screen.
The image of this linear transformation will be the plane $\Pi$ that you are projecting onto.  The kernel of this linear transformation can be used to answer the following question: when is the projection of two points the same?  (This corersponds to one object 'blocking' the view of the other on the screen.)  The answer is that $\vec{v}$ and $\vec{w}$ are projected to the same point in the plane if and only if $\vec{v} - \vec{w}$ lies in kernel of $\pi$.  In this sense, the kernel allows you to detect when two points are being projected onto the same point.
Linear transformations are more general than just this projection example, but the idea is still the same.  The image of a linear transformation $L : V \rightarrow W$ between two vector spaces just tells you the subspace of vectors in $W$ that are images of vectors in $V$ under the transformation $L$.  In other words, the image tells you where vectors get sent to.  The kernel of a linear transformation helps you detect when two vectors in $V$ are transformed to the same vector in $W$ under $L$.  That is, $L(\vec{v}) = L(\vec{v'})$ if and only if $\vec{v} - \vec{v'} \in \text{ker}(L)$.