Matrix Transformations (1/4): Domain, Codomain, Range, Standard Matrix [Passing Linear Algebra]

A matrix can be thought of as a tool to transform vectors. See video guide and some sweet bonus info below: Standard Matrix: 1:12 Example: 1:20 4 Most Common Types of Transformations: 5:21 Domain, Codomain, and Range: 6:47 The range of a transformation T(x) = Ax is the column space of A!! How could that be? Remember the range is all the outputs you can get by applying the transformation, and the transformation is just Ax. Ax is defined as a linear combination of the columns of A. If the range is all the possible outputs of Ax, it is all the possible linear combinations of the columns of A. That is, by definition, the span of the columns of A! Therefore range = col(A). Also, the example I gave in the video took vectors in R2 and transformed them into vectors still in R2. This is boring. Other (more fun) problems transform vectors into a whole nother dimension. This is possible if the standard matrix A is not square. If A is mxn, it takes in vectors in Rn and transforms them into vectors in Rm. Ponder that for a bit!