Linear Maps

Question 1

Basis

Answer 1

a set of ordered vectors which is a spanning set and linearly independent

Question 2

any v in a vector space can be uniquely written as:

Answer 2

v = λ₁e₁ + … + λₙeₙ

where e₁,…eₙ is a basis of V

Question 3

[|v|]col

Answer 3

(λ₁ .. λₙ) (in a column form)

where v = λ₁e₁ + … + λₙeₙ

Question 4

linear map

Answer 4

If V and W are vector spaces over 𝔽 then F is a linear map if
F(v+w) = F(v) + F(w)
F(λv) = λF(v)
where v,w are in V λ in 𝔽

Question 5

Lemma 4.3. Let V,W be a vector space over 𝔽 and F: v->W be a linear map. Then f(0) =

Answer 5

0

Question 6

best example of a linear map

Answer 6

matrix multiplication on the left F: v -> Av

Question 7

multiplying a matrix on the left with an element of the usual basis will

Answer 7

output the ith column of the matrix

Question 8

The matrix representation of a linear map F, [|F|]col, is given by

Answer 8

([|F(e₁)|], … [|F(eₙ)|])

Question 9

[|F|]col [|v|]col =

Answer 9

[|F(v)|]col

Question 10

How to apply a linear map where the basis is not standard

Answer 10

1. chose the basis
2. calculate F(eᵢ)
3. get F(eᵢ) into a L.C. of the basis elements
4. fill out [|F|]col = ([|f(e₁)|], … [|f(eₙ)|])
5. find [|v|]col w.r.t the basis
6. multiply [|F|]col [|v|]col to get the result

(to check sum all the basis elements = e and calculate F(e) )