Co-ordinates / Matrix representation of linear maps (BCM) Flashcards
proposition 7.1. scalars in a L.I. L.C.
if a set of n vectors from v is linearly independent then
λ₁x₁+… +λₙxₙ=μ₁x₁+…+μₙxₙ
means that λᵢ=μᵢ
[|v|]col
(λ₁ .. λₙ) (in a column form)
where v = λ₁e₁ + … + λₙeₙ
How to find the co-ordinates w.r.t basis
- write the vector as a linear combination of the basis
2. put into [|v|]col form
Theorem 7.4. ?
matrix representation of vectors in summation form.
Base change matrix from basis e₁,…,eₙ to basis e₁’,…,eₙ’
P = ([|e₁|]’col [|e₂|]’col … [|eₙ|]’col)
[|e|]’col = (λ₁ .. λₙ) where e = λ₁e₁’ + … + λₙeₙ’
so P is matrix filled with column representations of the original basis w.r.t the new basis.
How to use the base change matrix from e₁,…,eₙ to basis e₁’,…,eₙ’
[|v|]’col = P[|v|]col
the co-ordinates w.r.t. the old basis is the B.C.M. from old to new multiplied by the co-ordinates w.r.t the new basis.
matrix multiplication in summation form
Σⱼ₌ᵢⁿ Pᵢⱼλⱼ
i.e. the sum of the elements from 1-n of a coefficiant and the subsequent column of the matrix
B.C.M from A->B is equal to
A->E->E->B
where E is the usual basis
What is the B.C.M from E->B
just the matrix where columns are made up of the basis elements from B
Base change matrices are always…
invertible
if F:V->W. [|F(v)|]col =
w.r.t. ws
[|F|]col[|v|]col
w.r.t. vs
The matrix representation of a linear map F, [|F|]col, is given by
([|F(e₁)|], … [|F(eₙ)|])
Base change matrix summation (to get new basis element from old)
The indices are not adjacent
vᵢ’ =Σⱼ₌₁ⁿ Pⱼᵢvⱼ
[|F|]’col w.r.t. new basis is
P⁻¹AP
similar
A,B (both nxn matrices) are similar iff there is an invertible P s.t. B= P⁻¹AP
