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Brain dump on Vector Space
- linear independence:
- if 1 & 2, then 's define a basis (
- cardinality of basis is dimension
- if and are bases for , then .
Bases of Subspace
For a vector space with dimension less than ∞, if , then .
Let represent the space of polynomials. Then and
Column and Row Space
For a matrix, , where 's are columns.
is called the column space of .
is called row space of .
For vector space with dimension , then
- no set of fewer than vectors can span ;
- any subset of fewer than linearly independent vectors can be extended (i.e. vectors can be added) to form a basis for ; and
- any spanning set containing more than vectors can be pared down (i.e. vectors can be removed) to form a basis for .
Bases in Euclidean Space
is a basis iff , where is a matrix.
is standard basis in .
A standard basis for is a set such that th entry of is , and all other entries are
Let denote an ordered set representation, such that . will denote an ordered basis for , for example.
Conversion of Bases
To Standard Basis
For , , and let be a second basis (i.e. not standard).
We can use basis vectors to describe any point in .
- Given , find its coordinates w.r.t.
- Given , find its coordinates w.r.t. .
Case 2: .
Case 1: .
Note: The 2×2 matrix is called the transition matrix.
To Arbitrary Basis
Let and . Then (assuming and are nonsingular)
The product is the tranformation matrix from to
Cases for transformation matrix
Note that for standard basis is the identity matrix, so
For vector space with dimension , let be the ordered basis for . Thus .
The vector is called the coordinate vector of w.r.t. and is denoted .
Therefore, the previous section could be rewritten as
where and are ordered bases for and , respectively, and
Maximum number of points = 50 + 7 pt. bonus question
Look over Determinants, null spaces, and solving linear systems of equations.