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> What are orthogonal vectors good for? Any set of n linearly independent vectors B_a={\vec{a}_i}_{i=1..n} in a vector space can be used as a coordinate system
by ivan_ah 13y ago
> What are orthogonal vectors good for?
Any set of n linearly independent vectors B_a={\vec{a}_i}_{i=1..n} in a vector space can be used as a coordinate system, but the "computational cost" of finding the coordinates w.r.t. the basis B_a will be annoying. Each time you want to find the coordinate of a vector you have solve a system of linear equations.
A basis consisting of orthogonal vectors {\vec{e}_i}_{i=1..n} is way cooler because you can calculate the coefficients of any vector using the formula $v_i = (\vec{v} · \vec{e}_i)/||\vec{e}_i||²
Of course the best thing is to have an orthonormal basis B_s={\hat{e}_i}_{i=1..n}, so that the coefficients of a vector w.r.t B_s can be calculated simply v_i = \vec{v} · \hat{e}_i.
A projection onto the \hat{e}_i subspace.
> Why would we ever want a basis besides the standard basis in real Euclidean space?
Hmm... I'm thinking eigenbases? The operations of a matrix A for vectors expressed in terms of its eigenbasis is just a scaling, i.e. {B_e}_[A]_{B_e} = Q^{-1} {B_s}_[A]_{B_s} Q = diagonal matrix = Λ.
> Is Euclidean space the only vector space out there?
> Do vectors have to be lists of numbers?
Good point.
If I bump this to 5-6 pages, I will try to include something about generalized vector spaces.
Orthogonal polynomials could be a good one to cover.
- weaksauce 13y agomaybe bump it to two different handouts?
- algorias 13y ago> Each time you want to find the coordinate of a vector you have solve a system of linear equations. This is wrong, unless I'm misreading you. The first time you want the coordinates of a vector in a nonstandard basis you need to construct the matrix that does the change of basis, which requires inverting a matrix (same cost as solving a LSE), but after that, it's just plain matrix-vector multiplication.
- dahart 13y agoComputer graphics is full of fun topics that answer these questions in accessible and visible/tangible ways! Skeletal character rigging in particular motivates a nice understanding of orthonormal bases and why a basis other than the identity matrix is very useful. Even simple 2d conversions between screen space & pixel space, for example, can be useful motivational examples- its the thing your web browser did to render what you're reading. :) Not that anyone would want to cover those topics in a 4 page linear algebra primer, but maybe there is some inspiration there for ways to include pictures that explain, instead of more math notation... ;) The last 2 questions there are interesting and worthwhile topics for the interested student, but I'd say, just my $0.02, don't ruin a good thing by stuffing too much into it...
- j2kun 13y agoI knew the answers to these questions :) I'm just saying they belong in a discussion of the topics. Maybe I just like to take math slower than a 4-page summary.