You can think of a linear transformation as just a bunch of dot products. For example, here is a matrix describing a linear transformation that rotates vectors on a two-dimensional plane:

(cos(phi) sin(phi))
(-sin(phi) cos(phi))

When you multiply a two-dimensional column vector with this matrix, you take two dot-products, one with the upper row and one with the lower row. The first row of this matrix is an

unit vector, which makes angle phi with the x-axis. The second row is

orthogonal to the first row, which is easy to check by taking dot products. So if you think about it a bit, you see that this transformation must be either a rotation or a combined rotation and 'mirroring' of the vecto r around either axis. If you do some more thinking you can convince yourself that this particular transformation does not 'mirror' the vector, unlike this matrix, which mirrors a vector with respect to the x-axis:

(-1 0)
(0 1)