trying to be a mile wide AND a mile deep
The key to the equations of lines and planes in three dimensions is that, in each case, we need a point to locate the object in space, and a vector to tilt it at the correct angle. In each case, however, the kind of vector that unambiguously gives the direction of the object is different. […]
1. There is no one multiplication for vectors. You can define multiplication-like operations; some give scalars (dot product and other inner products) and some give vectors (cross product). Nicely, these behave like regular products when it comes to vector-valued functions: the product rule applies when you differentiate (though you must maintain ordering with cross product!). […]
Linear independence, spanning, and the three bears Papa’s bed is too hard – linearly independent, but too spare to span. Mama’s bed is too soft – spans, but too many options to suffocate in. Baby’s bed is just right – enough to span and nothing more. A basis! Coordinate vectors and sweets A restaurant serves […]
At the center of almost any proof involving linear transformations is to apply linearity to move between the domain and codomain, preserving the structure of linear combinations. Statements proved in such a way: all linear transformations take 0 to 0 images of subspaces are subspaces (special example: image) preimages of subspaces are subspaces (special example: […]
Which of the following are vector spaces over the scalar field R of real numbers? A) U = (R x R, +U, *U) where (a1, a2) +U (b1, b2) = (a1 + 2b1, a2 + 3b2) and *U is the usual scalar multiplication B) V = (R x R, +V, *V) where (a1, a2) +V […]
Parametrization is applied when an object is “really” of lower dimension than the space it lives in. Curves, whether they live in 2D or 3D, are really only one-dimensional. A surface, which might live in 3D (such as a sphere), is really only two-dimensional. The “really” is made rigorous by our ability to represent such […]