Column Space

Column space refers to all possible outputs you can get when you multiply a matrix by different vectors. Think of a matrix as a set of building blocks that, when combined with various inputs, produce different results. The column space is the collection of all these possible results—it's the "reach" or "span" of the matrix in space. If a vector lies within this space, it can be generated by some combination of the matrix's columns. Essentially, the column space shows what outputs are achievable using the matrix's columns as building blocks.