Prof Wildberger you are a fantastic teacher. Praying Tanaka has a final very soon and was struggling with this topic, but your conceptual explanation and the example were really helpful. Praying Tanaka hopes to dig through your lectures and find one on orthogonality too- another topic that didn't really click.
So the reason the nullity/dimension of the kernal is ONE (and the subspace is a line) is the occurence of THREE pivot columns in RREF, leaving a SINGLE parameter.
+Pauls SecondChannel Yes. From Row Echelon Form, you get to choose the variable corresponding to non-leading columns: in this case x_3. All the other values x_1,x_2 and x_4 are determined by the chosen value of x_3. The dimension comes from your freedom to choose x_3. Since there is only one degree of freedom here, the kernel is one dimensional.
Why is it R^4 at 0:50 ? The dimension of the environment isn't given by the lines of the matrix (which represent the coordinates of the vector)? The columns represent the no of vectors, not the coordinates which give the environment's dimension
