Linear Transformations Flashcards
Linear Transformation
Synonyms
- linear mapping
- linear map
- linear operator
- homomorphism
Linear Transformation
Definition
- let V, W be vector spaces over F, a map T:V->W is called a linear transformation if:
i) T(|v + |v’) = T(|v) + T(|v’) , for all v,v’ϵV
ii) T(a|v) = aT(|v) , for all aϵF and vϵV
Elementary Properties of Linear Transformations
-if T is a linear mapping then:
1) T(|0) = |0, if T(|0) ≠|0 then T is not linear
2) T(|-v) = -T(|v)
3) T(a|v + b|v’) = aT(|v) + bT(|v’), linear transformations preserve linear combinarions
-more generally, if {v1,v2,…,vn}ϵV and {a1,a2,…,an}ϵF
T(Σai |vi ) = Σ[ai T(|vi) ]
where sums are between i=1 and i=n
Linear Transformations
Special Cases
i) The Identity Map: Idv : |vϵV -> |vϵV is a linear map V->W
ii) The Zero Map: Z : |vϵV -> |0ϵW
iii) mxn Matrix A Ta : |vϵF^n -> A|vϵF^m
for |v={v1,v2,…,vn}ϵF^n :
Ta(|v) = A |v
Abstract Examples of Linear Transformations
1) Differentiation
2) Integration
3) The Definite Integral
4) Transposition Maps (transforms a matrix to its transpose
5) Trace Map (gives the trace of a matrix, the sum of all elements in the main diagonal
Linear Transformations
Matrix Proposition
-given a linear map T:F^n->F^m, there exists a uniquely defined matrix A such that T=Ta:|vϵF->A |v ϵF^m
List of Example Linear Transformations in ℝ² (and ℝ^3)
- Rotation
- Reflection
- Rescalings (Enlargement)
Transformation Matrix for Rotation in ℝ² (and ℝ^3)
A = (cosθ -sinθ)
(sinθ cosθ)
Matrix of Linear Transformation
-let V, W be finite dimensional vector spaces
-linear maps T:F^n -> F^m are always given by matrices
-there exists a unique matrix A with T=Ta: |vϵF^n -> A|vϵF^m
-the same is true for linear maps T: V->W, but bases must be fixed for V & W, the matrices we get depend on these bases
-let T: V->W be a linear map
let S={v1,…,vn} be a basis of V, let R={w1,…,wn} be a basis of W
-the matrix of T with respect to the bases is the matrix
A=aij whose jth column is T(|vj)
so
T(|v1) = a11 |w1 + a21 |w2 + … + am1 |wm
T(|v2) = a12 |w1 + a22 |w2 + … + am2 |wm
….
T(|vn) = a1n |w1 + a2n |w2 + … + amn |wm
How do you find the matrix of a transformation T with respect to the canonical bases of V and W?
- take the vectors that form the basis of V
- apply the transformation T to them
- write this results as coordinate vectors using the basis of W
- these coordinate vectors form the columns of the matrix A
How do you apply a transformation using the matrix of transformation with respect to the canonical bases?
- let T: V -> W
- write the vector as coordinate vectors in terms of the basis of V
- multiply the coordinate vectors by the transformation matrix, A
- the resulting vector will give you the coordinate vector with respect to the basis of W
Kernel
Definition
-let T: V->W be a linear mapping between vector spaces
-the null space or kernel of T is:
ker T = {vϵV : T(v)=0w} ⊆ V
-the kernel of T is the vector space of all vectors v in V which become the zero vector in W when the transformation T is applied to them
Image
Definition
-let T: V->W be a linear mapping between vector spaces
-the image of T is:
im T = {T(v) : vϵV} ⊆ W
-the image of T is the vector space in W of all vectors that result from the transformation T being applied to all vectors v in V
Kernel
Matrix Definition
-if A is an mxn matrix and Ta : vϵF^n -> AvϵF^m , then:
ker Ta = {vϵF^n : Av=|0}
i.e. the kernel of A is the solution space for the system of linear equations given by A
Image
Matrix Definition
-if A is an mxn matrix and Ta : vϵF^n -> AvϵF^m , then:
im Ta = {Av : vϵF^n} = bϵF^m
i.e. the system of linear equations Av = b is consistent/has solutions
Kernel, Image and Subspace
- the kernel and image are both subspaces
- let T: V -> W be a linear map
- the kernel of T is a subspace of V and the image of T is a subspace of W
Nullity
Definition
-the nullity of T is dim (ker T), written:
n(T) = dim( ker(T) )
Rank
Definition
-the rank of T is dim(im(T)), written:
r(T) = dim( im(T) )
What are the source and target?
- the source is another name for the domain
- the target is the same as the codomain
Composition of Linear Maps
-if S:U->V and T:V->W are linear maps, then the composition:
T∘ S : U->W is linear too
Composition of Linear Maps with Matrices
-given an mxn matrix A and an nxl matrix B we have linear maps
Tb : F^l -> F^n where Tb(v)=Bv and Ta : F^n->F^m where Ta(v)=Av
-the composition is Ta∘ Tb = Tab
i.e. (Ta∘Tb)(v) = ABv
Transition Matrices From One Basis to Another
- let U, V and W be vector spaces, T:U->V and T’:V->W are linear maps
- consider bases S of U, S’ of V and S’’ of W
- let A(S’,S) be the matrix of T w.r.t. S and S’
- let A’(S’‘,S’) be the matrix of T’ w.r.t. S’ and S’’
- then with respect to S & S’’ the matrix B=B(S’‘,S) of (T’∘T) : U->W is B(S’‘,S) = A’(S’‘,S’) A(S’,S)
Transition Matrices - Identity Transformation
-if id:V->V is the identity map and S, S’ are bases of V
-the transition matrix, writing S’ in terms of S, is the matrix I(S,S’) of id with respect to S’ in the domain of id and S in the codomain of id
I(S,S’) = (I(S’,S))^(-1)
Transition Matrix With Respect to Other Matrices
Derivation
-let T:V->W be a linear map
-S and S’ are bases of V & R and R’ are bases of W
-if A=A(R,S) is the matrix of T with respect to S and R
and A’=A(R’,S’), then:
A(R’,S’) = I(R’,R) A(R,S) I(S,S’)
i.e. A’ = Q^(-1) A P
