Im a second semester engineering student and im a few weeks into a linear algebra class. I understand most of it, but my teacher has to work to speak english so she doesn't explain things very well. We just started linear transformations and a few things seem unclear to me. Take a shear transformation for example. You multiply a vector by a matrix and end up with another vector - the 'image' of the original vector. There is an example in my book that considers the square formed by area under y=2 from x=0 to x=2. It considers every point as a vector, and then multiplies them by a matrix to get a transformation that looks like a parallelogram - the height of it is still two, but it is horizontally stretched. It shows calculations for a couple of the corner points. The first is the vector [0,2] ([x1,x2] - written vertical just hard to type that way :) ) multiplied by the 2x2 matrix ( [1,3] for the top row and [0,1] for the bottom ) to get [6,2] . Next the vector [2,2] is multiplied by the same matrix to get [8,2] . What does the matrix mean - what do the coefficients actually represent? If you consider a shape such as a square and you want to transform it to a parallelogram, how do you know what coefficients to use in the matrix?