rank nullity theorem

1. theorem

For vector spaces \(V\) and \(W\) and a linear mapping \(\Phi : V \rightarrow W\): \[ \dim(\ker(\Phi)) + \dim(\ker(\text{image}(\Phi))) = \dim(V) \]

the kernel is everything in \(V\) that gets sent to 0
to find the dimension of the kernel and the image, we are often going to be looking at the transformation matrix \(\mathbf{A}_{\Phi}\) for \(\Phi\).
- Remember that \(A_{\Phi}\) is given with respect to a specific coordinate system for \(V\) and \(W\) (see change of basis note)
\(\dim(\text{image}(\Phi)) = \text{rank}(\mathbf{A}_{\Phi})\). Remember that dimension is the number of vectors in a basis for a vector space (see rank).