concept standard basis vector in category python

This is an excerpt from Manning's book Math for Programmers: 3D graphics, machine learning, and simulations with Python MEAP V11. Login to get full access to this book.

Figure 4.32: The 2D vector (7, -4) as a linear combination of the standard basis vectors e1 and e2.

We’ve only written the same vectors in a slightly different way, but it turns out this change in perspective makes it easy to compute linear transformations. Because linear transformations respect linear combinations, all we need to know to compute a linear transformation is how it affects standard basis vectors.

The transformation expressed as “a rotation counterclockwise by 90° about the z-axis” can equivalently be described by what it does to the standard basis vectors e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1). Namely, the results are (0, 1, 0), (-1, 0, 0), and (0, 0, 1). Whether we think of this transformation geometrically or as described by these three vectors (or nine numbers), we’re thinking of the same imaginary machine (figure 5.1) that operates on 3D vectors. The implementations might be different, but the machines still produce indistinguishable results.

This is because all of the standard basis vectors are transformed to vectors with positive coordinates: (2, 1, 1), (1, 2, 1), and (1, 1, 2), respectively.

How the linear transformation defined by this matrix affects the standard basis vectors.