Linear Maps

Question 1

What is the domain and codomain of a linear map?

Answer 1

If T: V -> W
V is domain space (start)
W is codomain (end)

Question 2

How is the action of the linear map notated?

Answer 2

For a vector v in V:
T(v) is the application of map T onto v, where T(v) is a vector in W.

Question 3

What is the characteristics of the linear map?

Answer 3

Preserves addition and scalar multiplication. This means that doing these operations in any order will yield the same result.

Question 4

How is a linear map defined on a basis?

Answer 4

• Know how LM acts on basis vecotr -> extend to operate on entire space since basis used to build any vector
• Let T: V -> W. Then T(vi) = wi for all i.
• Write T(v) = T(linear combination of v). Apply scalar multiplication rule and substitute T(vi) with T(wi)

Question 5

How do we find the linear map of arbitrary vector given basis and linear maps?

Answer 5

V = linear comb of basis
Take linear map of linear comb
notice that T(v) = xT(e1) + yT(e2)

Question 6

What is the determinant of a linear operator?

Answer 6

On finite dimension VS:
det(T) := det([T]B<- B)

Question 7

What is the composition between two linear maps

Answer 7

• S: U -> V
• T: V-> W
  (S∘T)(u) := T(S(u)) where u in U

Question 9

How do we compute [T(v)] with respect to C for all vectors v in V?

Answer 9

[T(v)]C = COB matrix from B to C x coordinate vector of v wrt basis B

Question 10

How do we show that the coordinate vector of T(v) with respect to C is the multiplication of the COB matrix with the coordinate vector ito B

Answer 10

• Draw composition map
• Notice that the composition function between map that takes coordinate B -> V, then V -> W, then W -> coordinate C is multiplication of COB any u in space with coordiante vectors of B

Question 11

What is the inverse of a matrix of a linear map?

Answer 11

The matrix of the inverse of the linear map with bases switched around
e.g [T]C<-B = [T-1]B <- C

Question 12

How do we form a composition matrix between 2 linear maps?

Answer 12

• T:U-> V
• S:V-> W
• Given bases B, C, D for U,V,W

The COB of composition matrix (T∘S)D<-B = COB of T from C <- B x COB of S from D <- C

Question 13

What is the COB matrix for an m-dimensional vector space?

Answer 13

P from C <- B = COB matrix of identity matrix from C <- B

Question 14

How do we calculate the matrix for a linear map with respect to a different basis given the matrix w.r.t one basis and a change of basis matrix?

Answer 14

• B and C bases for V
• T: V->W
• Let P = COB matrix for identity matrix from C<-B
  COB of T from C to C = invserse of P x COB of T from B<-B x P