Definition, basic operations
A sequence of n elements of \(f\), v, is a vector.
Addition, scalar multiplication
Addition of vectors is naturally defined: \((u + v)_i = u_i + v_i\). Scalar multiplication is similarly defined.
Geometric model: coefficient sequence
Can be interpreted as a point in the cartesian space \(F^{n}\); but which point? That depends on what the basis vectors model, the units, the inner product between them.
As a combination of standard basis vectors
Define standard basis vectors: \(e_i\) as a vector with 1 in the ith position, 0 elesewhere.
So, v can be viewed as a combination \(v_i e_i\) of the standard basis vectors \(\set{e_i}\).
Standard basis models what?
The standard basis are usually considered with the standard inner product: \(\dprod{x, y} = \sum_i x_i y_i\). But this need not be the case. Other bases are possible.
Equivalent representations of the same point
So, the vector representing a point in a coordinate space changes, when the basis chosen is different. Change of basis operation is a linear operation, for details see linear algebra ref.
Combinations of vectors
Linear combination
\(\sum a_{i} v_{i} = p\).
Conic/ non-ve combination
Coefficients \(a_{i} \geq 0\).
Affine combination
If coefficients \(\sum a_{i} = 1\), this is an affine combination.
Colleniearity preserved
Make affine combo \(p = ax + (1-a)y\); take vector \(x - p = (1-a)(x-y)\); this has same inner product with the basis vectors as x-y.
Convex combination
Affine combo where \(a_{i} \geq 0\): both affine and conic.