Some of these points about matrices are worth noting down, as aids to intuition. I might expand on some of these points into their own posts.
Note that the above diagram is not mathematically correct. I drew 5 basis vectors in 2D space, and you cannot have more than 2 linearly independent basis vectors in two dimensions. This diagram is simply for illustration purposes.
The determinant of a matrix is essentially the volume spanned by the basis vectors formed by its columns. A degenerate matrix has a determinant of zero because the measurement of this “hypervolume” on one axis becomes zero.
The left null space of a matrix represents the set of normal vectors for the hyperplane defined by the column space of this matrix. This is because by definition, all vectors in the left null space are orthogonal to the column space. Thus, any vector in the left null space also represents the actual geometric equation of the hyperplane defined by the column space.