Q: can we say every linear transformation (linT) can be /characterized/expressed/represented/ as a multiplication by a specific (often square) matrix? Yes See P168 [[the manga guide to LT]] BTW, The converse is easier to prove — every multiplication by a matrix is a linT, assuming input is a columnar vector. Before we can learn the practical techniques applying MxV on LinT, we have to clear a lot of abstract and confusing points. LinT is one of the more abstract topics. 1) What kind of inputs go into a LinT? By LinT definition, real numbers can qualify as input to a LinT. With this kinda input, a LinT is nothing but a linear function of the input variable x. Both the Domain and the Range of the linT consist of real numbers.2) This kinda linear transform is too simple, not too useful, kinda degenerate. The kinda input we are more interested in are vectors, expressed as columnar vectors. With this kinda inputs, each LinT is represented as a matrix. A simple example is a “scaling” where input is a 3D vector (x,y,z). You can also say every point in the 3D Space enters this LinT and “maps” to a point in another 3D space. This transform specifies how to map Every single point in the input space. “Any point in the 3D space I know exactly how to map!”. Actually this is a kind of math Function. Actually Function is a fundamental concept in Linear Transformation. This particular transform doesn’t restrict what value of x,y or z can come in. However, the parameters of the function itself is locked down and very specific. This is a specific Function and a specific Mapping.3) Now, since matrix multiplication can happen between 2 matrices, so what if input is a matrix? Will it be a LinT? I don’t know too much but I feel this is not practically useful. The most useful and important kind of Linear transformation is the MxV.4) So what other inputs can a LinT have? I don’t know. To recap, there are unlimited types of linear transformations, and each LinT has an unlimited, unconstrained Domain. This makes LinT a a rather abstract topic. We must divide and conquer. First divide the “world of linear transforms” by the type of input. The really important type of input is columnar vector. Once we limit ourselves to columnars, we realize every LinT can be written as a LHS multiplying matrix. To get a concrete idea of LinT, we can start with the 2D space — so all the input columnars come from this space. These can be represented as points in the 2D space.